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Discrete Mathematics and Symmetry Edited by
Angel Garrido Printed Edition of the Special Issue Published in Symmetry
www.mdpi.com/journal/symmetry
Discrete Mathematics and Symmetry
Discrete Mathematics and Symmetry
Special Issue Editor Angel Garrido
MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade
Special Issue Editor Angel Garrido Department of Fundamental Mathematics, Faculty of Sciences, UNED Spain
Editorial Ofﬁce MDPI St. AlbanAnlage 66 4052 Basel, Switzerland
This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 20738994) from 2018 to 2020 (available at: https://www.mdpi.com/journal/symmetry/ special issues/Discrete Mathematics Symmetry).
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Contents About the Special Issue Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Preface to ”Discrete Mathematics and Symmetry” . . . . . . . . . . . . . . . . . . . . . . . . . .
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Michal Staˇs On the Crossing Numbers of the Joining of a Speciﬁc Graph on Six Vertices with the Discrete Graph Reprinted from: Symmetry 2020, 12, 135, doi:10.3390/sym12010135 . . . . . . . . . . . . . . . . .
1
Abdollah Alhevaz, Maryam Baghipur, Hilal Ahmad Ganie and Yilun Shang Bounds for the Generalized Distance Eigenvalues of a Graph Reprinted from: Symmetry 2019, 11, 1529, doi:10.3390/sym11121529 . . . . . . . . . . . . . . . . . 13 Jelena Daki´c, Sanja Janˇci´cRaˇsovi´c and Irina Cristea Weak Embeddable HypernearRings Reprinted from: Symmetry 2019, 11, 964, doi:10.3390/sym11080964 . . . . . . . . . . . . . . . . . 30 Ahmed A. Elsonbaty and Salama Nagy Daoud Edge Even Graceful Labeling of Cylinder Grid Graph Reprinted from: Symmetry 2019, 11, 584, doi:10.3390/sym11040584 . . . . . . . . . . . . . . . . . 40 Xiaohong Zhang and Xiaoying Wu Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups Reprinted from: Symmetry 2019, 11, 553, doi:10.3390/sym11040553 . . . . . . . . . . . . . . . . . 70 Yilun Shang Isoperimetric Numbers of Randomly Perturbed Intersection Graphs Reprinted from: Symmetry 2019, 11, 452, doi:10.3390/sym11040452 . . . . . . . . . . . . . . . . . 83 Shunyi Liu Generalized Permanental Polynomials of Graphs Reprinted from: Symmetry 2019, 11, 242, doi:10.3390/sym11020242 . . . . . . . . . . . . . . . . . 92 Sergei Sokolov, Anton Zhilenkov, Sergei Chernyi, Anatoliy Nyrkov and David Mamunts Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order Reprinted from: Symmetry 2019, 11, 236, doi:10.3390/sym11020236 . . . . . . . . . . . . . . . . . 104 Michal Staˇs Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five Reprinted from: Symmetry 2019, 11, 123, doi:10.3390/sym11020123 . . . . . . . . . . . . . . . . . 116 Salama Nagy Daoud Edge Even Graceful Labeling of Polar Grid Graphs Reprinted from: Symmetry 2019, 11, 38, doi:10.3390/sym11010038 . . . . . . . . . . . . . . . . . . 125 Xiujun Zhang, Xinling Wu, Shehnaz Akhter, Muhammad Kamran Jamil, JiaBao Liu, Mohammad Reza Farahani EdgeVersion AtomBond Connectivity and Geometric Arithmetic Indices of Generalized Bridge Molecular Graphs Reprinted from: Symmetry 2018, 10, 751, doi:10.3390/sym10120751 . . . . . . . . . . . . . . . . . 151 v
Lifeng Li, Qinjun Luo Sufﬁcient Conditions for Triangular Norms Preserving ⊗−Convexity Reprinted from: Symmetry 2018, 10, 729, doi:10.3390/sym10120729 . . . . . . . . . . . . . . . . . 167 Huilin Xu and Yuhui Xiao A Novel Edge Detection Method Based on the Regularized Laplacian Operation Reprinted from: Symmetry 2018, 10, 697, doi:10.3390/sym10120697 . . . . . . . . . . . . . . . . . 177 JiaBao Liu and Salama Nagy Daoud The Complexity of Some Classes of Pyramid Graphs Created from a Gear Graph Reprinted from: Symmetry 2018, 10, 689, doi:10.3390/sym10120689 . . . . . . . . . . . . . . . . . 186 ZhanAo Xue, DanJie Han, MinJie Lv and Min Zhang Novel ThreeWay Decisions Models with MultiGranulation Rough Intuitionistic Fuzzy Sets Reprinted from: Symmetry 2018, 10, 662, doi:10.3390/sym10110662 . . . . . . . . . . . . . . . . . 207 Firstname Lastname, Firstname Lastname and Firstname Lastname Maximum Detour–Harary Index for Some Graph Classes Reprinted from: Symmetry 2018, 10, 608, doi:10.3390/sym10110608 . . . . . . . . . . . . . . . . . 232 Xiaohong Zhang, Rajab Ali Borzooei and Young Bae Jun QFilters of Quantum BAlgebras and Basic Implication Algebras Reprinted from: Symmetry 2018, 10, 573, doi:10.3390/sym10110573 . . . . . . . . . . . . . . . . . 244 Aykut Emniyet and Memet S¸ahin Fuzzy Normed Rings Reprinted from: Symmetry 2018, 10, 515, doi:10.3390/sym10100515 . . . . . . . . . . . . . . . . . 258 Fabian Ball and Andreas GeyerSchulz Invariant Graph Partition Comparison Measures Reprinted from: Symmetry 2018, 10, 504, doi:10.3390/sym10100504 . . . . . . . . . . . . . . . . . 266 Yanlan Mei, Yingying Liang and Yan Tu A MultiGranularity 2Tuple QFD Method and Application to Emergency Routes Evaluation Reprinted from: Symmetry 2018, 10, 484, doi:10.3390/sym10100484 . . . . . . . . . . . . . . . . . 290 Xiaoying Wu and Xiaohong Zhang The Structure Theorems of PseudoBCI Algebras in Which Every Element is QuasiMaximal Reprinted from: Symmetry 2018, 10, 465, doi:10.3390/sym10100465 . . . . . . . . . . . . . . . . . 306 Qing Yang, Zengtai You and Xinshang You A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons Reprinted from: Symmetry 2018, 10, 447, doi:10.3390/sym10100447 . . . . . . . . . . . . . . . . . 319 Zhanao Xue, Minjie Lv, Danjie Han and Xianwei Xin MultiGranulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation Reprinted from: Symmetry 2018, 10, 446, doi:10.3390/sym10100446 . . . . . . . . . . . . . . . . . 336 Jingqian Wang, Xiaohong Zhang Four Operators of Rough Sets Generalized to Matroids and a Matroidal Method for Attribute Reduction Reprinted from: Symmetry 2018, 10, 418, doi:10.3390/sym10090418 . . . . . . . . . . . . . . . . . 360 vi
Hu Zhao and HongYing Zhang Some Results on Multigranulation Neutrosophic Rough Sets on a Single Domain Reprinted from: Symmetry 2018, 10, 417, doi:10.3390/sym10090417 . . . . . . . . . . . . . . . . . 376 Mobeen Munir, Asim Naseem, Akhtar Rasool, Muhammad Shoaib Saleem and Shin Min Kang Fixed Points Results in Algebras of Split Quaternion and Octonion Reprinted from: Symmetry 2018, 10, 405, doi:10.3390/sym10090405 . . . . . . . . . . . . . . . . . 388 Jianping Wu, Boliang Lin, Hui Wang, Xuhui Zhang, Zhongkai Wang and Jiaxi Wang Optimizing the HighLevel Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modiﬁed Particle Swarm Optimization Algorithm Reprinted from: Symmetry 2018, 10, 349, doi:10.3390/sym10080349 . . . . . . . . . . . . . . . . . 406 Jihyun Choi and JaeHyouk Lee Binary Icosahedral Group and 600Cell Reprinted from: Symmetry 2018, 10, 326, doi:10.3390/sym10080326 . . . . . . . . . . . . . . . . . 420 Marija Maksimovi´c Enumeration of Strongly Regular Graphs on up to 50 Vertices Having S3 as an Automorphism Group Reprinted from: Symmetry 2018, 10, 212, doi:10.3390/sym10060212 . . . . . . . . . . . . . . . . . 434
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About the Special Issue Editor Angel Garrido, Ch. of Differential Geometry, Department of Fundamental Mathematics, Faculty of ´ ´ Sciences, UNED, Madrid, Spain. BIOGRAPHYFull Name: Angel Laureano Garrido BullonCurrent position: Chairman of Mathematics. Permanent Professor Doctor of the Faculty of Sciences of the UNED. Department of Fundamental Mathematics.Professional Experience: Polytechnic University of Madrid; University of Manchester; UNED.Degree: Licenciado (Degree) in Exact Sciences in the Faculty of Sciences of the Complutense University of Madrid. Programmer and Analyst at IBM. Master in Artiﬁcial Intelligence by the UNED. Full PhD studies in Mathematics, Informatics and Philosophy. PhD in Philosophy (UNED), especially of Logic and Foundations, with Summa Cum Laude in Thesis, by unanimity, and First Extraordinary Prize.He is the author of 27 books, published in prestigious editorials. Editorin Chief of Axioms journal. MDPI Foundation Verlag, Basel, Switzerland. 232 papers published. Editor of Symmetry and Education Sciences, MDPI, as well as Brain, Bacau University, amongst others. CoDirector of [email protected], a publication of the Faculty of Sciences, UNED, Madrid. Gold Medal from the University of Bacau.First Birkh¨auser Prize, ICM, 2016.
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Preface to ”Discrete Mathematics and Symmetry” SYMMETRY AND GEOMETRY. One of the core concepts essential to understanding natural phenomena and the dynamics of social systems is the concept of “relation”. Furthermore, scientists rely on relational structures with high levels of symmetry because of their optimal behavior and high performance. Human friendships, social and interconnection networks, trafﬁc systems, chemical structures, etc., can be expressed as relational structures. A mathematical model capturing the essence of this situation is a combinatorial object exhibiting a high level of symmetry, and the underlying mathematical discipline is algebraic combinatorics—the most vivid expression of the concept of symmetry in discrete mathematics. The purpose of this Special Issue of the journal Symmetry is to present some recent developments as well as possible future directions in algebraic combinatorics. Special emphasis is given to the concept of symmetry in graphs, ﬁnite geometries, and designs. Of interest are solutions of longstanding open problems in algebraic combinatorics, as well as contributions opening up new research topics encompassing symmetry within the boundaries of discrete mathematics but with the possibility of transcending these boundaries. Prof. Dr. Angel Garrido, Guest Editor. Angel Garrido Special Issue Editor
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SS symmetry Article
On the Crossing Numbers of the Joining of a Speciﬁc Graph on Six Vertices with the Discrete Graph Michal Staš Faculty of Electrical Engineering and Informatics, Technical University of Košice, 042 00 Košice, Slovakia; [email protected] Received: 19 December 2019; Accepted: 19 December 2019; Published: 9 January 2020
Abstract: In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G ∗ + Dn for the graph G ∗ on six vertices consists of one vertex which is adjacent with three nonconsecutive vertices of the 5cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G ∗ exactly once. These minimum symmetrical values are described in the individual symmetric tables. Keywords: graph; good drawing; crossing number; join product; cyclic permutation
1. Introduction An investigation on the crossing number of graphs is a classical and very difﬁcult problem. Garey and Johnson [1] proved that this problem is NPcomplete. Recall that the exact values of the crossing numbers are known for only a few families of graphs. The purpose of this article is to extend the known results concerning this topic. In this article, we use the deﬁnitions and notation of the crossing numbers of graphs presented by Klešˇc in [2]. Kulli and Muddebihal [3] described the characterization for all pairs of graphs which join product of a planar graph. In the paper, some parts of proofs are also based on Kleitman’s result [4] on the crossing numbers for some complete bipartite graphs. More precisely, he showed that cr(Km,n ) =
m m − 1 n n − 1 2
2
2
2
,
for
m ≤ 6.
Again, by Kleitman’s result [4], the crossing numbers for the join of two different paths, the join of two different cycles, and also for the join of path and cycle, were established in [2]. Further, the exact values for crossing numbers of G + Dn and of G + Pn for all graphs G on less than ﬁve vertices were determined in [5]. At present, the crossing numbers of the graphs G + Dn are known only for few graphs G of order six in [6–9]. In all these cases, the graph G is usually connected and includes at least one cycle. The methods in the paper mostly use the combinatorial properties of cyclic permutations. For the ﬁrst time, the idea of conﬁgurations is converted from the family of subgraphs which do not cross the edges of the graph G ∗ of order six onto the family of subgraphs whose edges cross the edges of G ∗ just once. According to this algebraic topological approach, we can extend known results for the crossing numbers of new graphs. Some of the ideas and methods were used for the ﬁrst time in [10]. In [6,8,9], some parts of proofs were done with the help of software which is described in detail in [11]. It is important to recall that the methods presented in [5,7,12] do not sufﬁce to determine the crossing number of the graph G ∗ + Dn . Also in this article, some parts of proofs can be simpliﬁed by utilizing the work of the software that generates all cyclic permutations in [11]. Its C++ version is located also on the website http://web.tuke.sk/feikm/coga/, and the list with all short names of 120 cyclic permutations of six elements have already been collected in Table 1 of [8]. Symmetry 2020, 12, 135; doi:10.3390/sym12010135
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Symmetry 2020, 12, 135
2. Cyclic Permutations and Corresponding Conﬁgurations of Subgraphs Let G ∗ be the connected graph on six vertices consisting of one vertex which is adjacent with three nonconsecutive vertices of the 5cycle. We consider the join product of the graph G ∗ with the discrete graph Dn on n vertices. It is not difﬁcult to see that the graph G ∗ + Dn consists of just one copy of the graph G ∗ and of n vertices t1 , . . . , tn , where any vertex t j , j = 1, . . . , n, is adjacent to every vertex of the graph G ∗ . Let T j , j = 1, . . . , n, denote the subgraph which is uniquely induced by the six edges incident with the ﬁxed vertex t j . This means that the graph T 1 ∪ · · · ∪ T n is isomorphic with K6,n and G ∗ + Dn = G ∗ ∪ K6,n = G ∗ ∪
n
Tj .
(1)
j =1
In the paper, the definitions and notation of the cyclic permutations and of the corresponding configurations of subgraphs for a good drawing D of the graph G∗ + Dn presented in [8] are used. The rotation rotD (t j ) of a vertex t j in the drawing D is the cyclic permutation that records the (cyclic) counterclockwise order in which the edges leave t j , see [10]. We use the notation (123456) if the counterclockwise order of the edges incident with the vertex t j is t j v1 , t j v2 , t j v3 , t j v4 , t j v5 , and t j v6 . Recall that a rotation is a cyclic permutation. Moreover, as we have already mentioned, we separate all subgraphs T j , j = 1, . . . , n, of the graph G∗ + Dn into three mutuallydisjoint families depending on how many times the edges of G∗ are crossed by the edges of the considered subgraph T j in D. This means, for j = 1, . . . , n, let R D = { T j : crD (G∗ , T j ) = 0} and SD = { T j : crD (G∗ , T j ) = 1}. The edges of G∗ are crossed by each other subgraph T j at least twice in D. For T j ∈ R D ∪ SD , let F j denote the subgraph G∗ ∪ T j , j ∈ {1, 2, . . . , n}, of G∗ + Dn , and let D( F j ) be its subdrawing induced by D. If we would like to obtain an optimal drawing D of G ∗ + Dn , then the set R D ∪ SD must be nonempty provided by the arguments in Theorem 1. Thus, we only consider drawings of the graph G ∗ for which there is a possibility of obtaining a subgraph T j ∈ R D ∪ SD . Since the graph G ∗ contains the 6cycle as a subgraph (for brevity, we can write C6 ( G ∗ )), we have to assume only crossings between possible subdrawings of the subgraph C6 ( G ∗ ) and two remaining edges of G ∗ . Of course, the edges of the cycle C6 ( G ∗ ) can cross themselves in the considered subdrawings. The vertex notation of G ∗ will be substantiated later in all drawings in Figure 1. First, assume a good drawing D of G ∗ + Dn in which the edges of G ∗ do not cross each other. In this case, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1a. Clearly, the set R D is empty. Our aim is to list all possible rotations rotD (t j ) which can appear in D if the edges of G ∗ are crossed by the edges of T j just once. There is only one possible subdrawing of F j \ {v4 } represented by the rotation (16532), which yields that there are exactly ﬁve ways of obtaining the subdrawing of G ∪ T j depending on which edge of the graph G ∗ can be crossed by the edge t j v4 . We denote these ﬁve possibilities by Ak , for k = 1, . . . , 5. For our considerations over the number of crossings of G ∗ + Dn , it does not play a role in which of the regions is unbounded. So we can assume the drawings shown in Figure 2. Thus, the conﬁgurations A1 , A2 , A3 , A4 , and A5 are represented by the cyclic permutations (165324), (165432), (146532), (165342), and (164532), respectively. Of course, in a ﬁxed drawing of the graph G ∗ + Dn , some conﬁgurations from M = {A1 , A2 , A3 , A4 , A5 } need not appear. We denote by M D the set of all conﬁgurations that exist in the drawing D belonging to the set M.
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Figure 1. Six possible drawings of G ∗ with no crossing among edges of C6 ( G ∗ ). (a): the planar drawing of G ∗ ; (b): the drawing of G ∗ with crD ( G ∗ ) = 1 and without crossing on edges of C6 ( G ∗ ); (c): the drawing of G ∗ only with two crossings on edges of C6 ( G ∗ ); (d): the drawing of G ∗ with crD ( G ∗ ) = 2 and with one crossing on edges of C6 ( G ∗ ); (e): the drawing of G ∗ only with one crossing on edges of C6 ( G ∗ ); (f): the drawing of G ∗ with crD ( G ∗ ) = 2 and with one crossing on edges of C6 ( G ∗ ).
Figure 2. Drawings of ﬁve possible conﬁgurations from M of the subgraph
Fj.
Recall that we are able to extend the idea of establishing minimum values of crossings between two different subgraphs onto the family of subgraphs which cross the edges of G ∗ exactly once. Let X and Y be the conﬁgurations from M D . We denote by crD (X and Y ) the number of crossings in D between T i and T j for different T i , T j ∈ SD such that Fi and F j have conﬁgurations X and Y , respectively. Finally, let cr(X , Y ) = min{crD (X , Y )} over all possible good drawings of G ∗ + Dn with X , Y ∈ M D .
3
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Our aim is to determine cr(X , Y ) for all such pairs X , Y ∈ M. In particular, the conﬁgurations A1 and A2 are represented by the cyclic permutations (165324) and (165432), respectively. Since the minimum number of interchanges of adjacent elements of (165324) required to produce cyclic permutation (165432) is two, we need at least four interchanges of adjacent elements of (165432) to produce cyclic permutation (165324) = (142356). (Let T x and T y be two different subgraphs represented by their rot(t x ) and rot(ty ) of length m, m ≥ 3. If the minimum number of interchanges of adjacent elements of 1 rot(t x ) required to produce rot(ty ) is at most z, then crD ( T x , T y ) ≥ m2 m− − z. Details have been 2 worked out by Woodall [13].) So any subgraph T j with the conﬁguration A2 of F j crosses the edges of T i with the conﬁguration A1 of Fi at least four times; that is, cr(A1 , A2 ) ≥ 4. The same reasoning gives cr(A1 , A3 ) ≥ 5, cr(A1 , A4 ) ≥ 5, cr(A1 , A5 ) ≥ 4, cr(A2 , A3 ) ≥ 4, cr(A2 , A4 ) ≥ 5, cr(A2 , A5 ) ≥ 5, cr(A3 , A4 ) ≥ 4, cr(A3 , A5 ) ≥ 5, and cr(A4 , A5 ) ≥ 4. Clearly, also cr(Ai , Ai ) ≥ 6 for any i = 1, . . . , 5. All resulting lower bounds for the number of crossings of two conﬁgurations from M are summarized in the symmetric Table 1 (here, Ak and Al are conﬁgurations of the subgraphs Fi and F j , where k, l ∈ {1, 2, 3, 4, 5}). Table 1. The necessary number of crossings between T i and T j for the conﬁgurations Ak , Al . 
A1
A2
A3
A4
A5
A1 A2 A3 A4 A5
6 4 5 5 4
4 6 4 5 5
5 4 6 4 5
5 5 4 6 4
4 5 5 4 6
Assume a good drawing D of the graph G ∗ + Dn with just one crossing among edges of the graph (in which there is a possibility of obtaining of subgraph T j ∈ R D ∪ SD ). At ﬁrst, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1b. Of course, the set R D can be nonempty, but our aim will be also to list all possible rotations rotD (t j ) which can appear in D if the edges of G ∗ are crossed by the edges of T j just once. Since the edges v1 v2 , v2 v3 , v1 v6 , and v5 v6 of G ∗ can be crossed by the edges t j v3 , t j v1 , t j v5 , and t j v1 , respectively, these four ways under our consideration can be denoted by Bk , for k = 1, 2, 3, 4. Based on the aforementioned arguments, we assume the drawings shown in Figure 3. Thus, the conﬁgurations B1 , B2 , B3 , and B4 are uniquely represented by the cyclic permutations (165423), (126543), (156432), and (154326), respectively. Because some conﬁgurations from N = {B1 , B2 , B3 , B4 } may not appear in a ﬁxed drawing of G ∗ + Dn , we denote by N D the subset of N consisting of all conﬁgurations that exist in the drawing D. Further, due to the properties of the cyclic rotations, we can easily verify that cr(Bi , B j ) ≥ 4 for any i, j ∈ {1, 2, 3, 4}, i = j. (Let us note that this idea was used for an establishing the values in Table 1) In addition, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1e. In this case, the set R D is also empty. Hence, our aim is to list again all possible rotations rotD (t j ) which can appear in D if T j ∈ SD . Since there is only one subdrawing of F j \ {v3 } represented by the rotation (16542), there are four ways to obtain the subdrawing of F j depending on which edge of G ∗ is crossed by the edge t j v3 . These four possibilities under our consideration are denoted by Ek , for k = 1, 2, 3, 4. Again, based on the aforementioned arguments, we assume the drawings shown in Figure 4. G∗
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Figure 3. Drawings of four possible conﬁgurations from N of the subgraph F j .
Figure 4. Drawings of four possible conﬁgurations from O of the subgraph F j .
Thus, the configurations E1 , E2 , E3 , and E4 are represented by the cyclic permutations (165432), (163542), (165342), and (136542), respectively. Again, we denote by OD the subset of O = {E1 , E2 , E3 , E4 } consisting of all conﬁgurations that exist in the drawing D. Further, due to the properties of the cyclic rotations, all lowerbounds of number of crossings of two conﬁgurations from O can be summarized in the symmetric Table 2 (here, Ek and El are conﬁgurations of the subgraphs Fi and F j , where k, l ∈ {1, 2, 3, 4}). 5
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Table 2. The necessary number of crossings between T i and T j for the conﬁgurations Ek , El . 
E1
E2
E3
E4
E1 E2 E3 E4
6 4 5 4
4 6 5 5
5 5 6 4
4 5 4 6
Finally, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1f. In this case, the set R D is also empty. So our aim will be to list again all possible rotations rotD (t j ) which can appear in D if T j ∈ SD . Since there is only one subdrawing of F j \ {v2 } represented by the rotation (16543), there are three ways to obtain the subdrawing of F j depending on which edge of G ∗ is crossed by the edge t j v2 . These three possibilities under our consideration are denoted by Fk , for k = 1, 2, 3. Again, based on the aforementioned arguments, we assume the drawings shown in Figure 5.
Figure 5. Drawings of three possible conﬁgurations from P of the subgraph F j .
Thus, the configurations F1 , F2 , and F3 are represented by the cyclic permutations (165432), (162543), and (126543), respectively. Again, we denote by P D the subset of P = {F1 , F2 , F3 } consisting of all configurations that exist in the drawing D. Further, due to the properties of the cyclic rotations, all lowerbounds of number of crossings of two configurations from P can be summarized in the symmetric Table 3 (here, Fk and Fl are configurations of the subgraphs Fi and F j , where k, l ∈ {1, 2, 3}). Table 3. The necessary number of crossings between T i and T j for the conﬁgurations Fk and Fl . 
F1
F2
F3
F1 F2 F3
6 4 5
4 6 5
5 5 6
3. The Crossing Number of G ∗ + Dn Recall that two vertices ti and t j of G ∗ + Dn are antipodal in a drawing D of G ∗ + Dn if the subgraphs T i and T j do not cross. A drawing is antipodalfree if it has no antipodal vertices. For easier and more accurate labeling in the proofs of assertions, let us deﬁne notation of regions in some subdrawings of G ∗ + Dn . The unique drawing of G ∗ as shown in Figure 1a contains four different regions. Let us denote these four regions by ω1,2,3,4 , ω1,4,5,6 , ω3,4,5 , and ω1,2,3,5,6 depending on which of vertices are located on the boundary of the corresponding region. Lemma 1. Let D be a good and antipodalfree drawing of G ∗ + Dn , for n > 3, with the drawing of G ∗ with the vertex notation like that in Figure 1a. If T u , T v , T t ∈ SD are three different subgraphs such that F u , F v , and F t have three different conﬁgurations from the set {Ai , A j , Ak } ⊆ M D with i + 2 ≡ j + 1 ≡ k (mod 5), then crD ( G ∗ ∪ T u ∪ T v ∪ T t , T m ) ≥ 6
for any T m ∈ SD . 6
Symmetry 2020, 12, 135
Proof of Lemma 1. Let us assume the conﬁgurations A1 of F u , A2 of F v , and A3 of F t . It is obvious that crD ( T u ∪ T v ∪ T t , T m ) ≥ 3 holds for any subgraph T m , m = u, v, t. Further, if crD ( G ∗ , T m ) > 2, then we obtain the desired result crD ( G ∗ ∪ T u ∪ T v ∪ T t , T m ) ≥ 3 + 3 = 6. To ﬁnish the proof, let us suppose that there is a subgraph T m ∈ SD such that T m crosses exactly once the edges of each subgraph T u , T v , and T t , and let also consider crD ( G ∗ , T m ) = 2. As crD ( T u , T m ) = 1, the vertex tm must be placed in the quadrangular region with four vertices of G ∗ on its boundary; that is, tm ∈ ω1,4,5,6 . Similarly, the assumption crD ( T t , T m ) = 1 enforces that tm ∈ ω1,2,3,4 . Since the vertex tm cannot be placed simultaneously in both regions, we obtain a contradiction. The proof proceeds in the similar way also for the remaining possible cases of the conﬁgurations of subgraphs F u , F v , and F t , and the proof is done. Now we are able to prove the main result of the article. We can calculate the exact values of crossing numbers for small graphs using an algorithm located on a website http://crossings.uos.de/. It uses an ILP formulation based on Kuratowski subgraphs. The system also generates veriﬁable formal proofs like those described in [14]. Unfortunately, the capacity of this system is limited. Lemma 2. cr( G ∗ + D1 ) = 1 and cr( G ∗ + D2 ) = 3. Theorem 1. cr( G ∗ + Dn ) = 6
n 2
n −1 2
+n+
n 2
for n ≥ 1.
1 + n + n2 Proof of Theorem 1. Figure 6 offers the drawing of G ∗ + Dn with exactly 6 n2 n− 2 n n−1 n ∗ crossings. Thus, cr( G + Dn ) ≤ 6 2 + n + 2 . We prove the reverse inequality by induction 2 on n. By Lemma 2, the result is true for n = 1 and n = 2. Now suppose that, for some n ≥ 3, there is a drawing D with n n − 1 n crD ( G ∗ + Dn ) < 6 +n+ (2) 2 2 2 and that cr( G ∗ + Dm ) ≥ 6
m m − 1 2
2
+m+
m
for any integer m < n.
2
(3)
Figure 6. The good drawing of G ∗ + Dn with 6
n n−1 2
2
+n+
n 2
crossings.
Let us ﬁrst show that the considered drawing D must be antipodalfree. For a contradiction, suppose, without loss of generality, that crD ( T n−1 , T n ) = 0. If at least one of T n−1 and T n , say T n , does not cross G ∗ , it is not difﬁcult to verify in Figure 1 that T n−1 must cross G ∗ ∪ T n at least trice; that is, crD ( G ∗ , T n−1 ∪ T n ) ≥ 3. From [4], we already know that cr(K6,3 ) = 6, which yields that the edges of the subgraph T n−1 ∪ T n are crossed by any T k , k = 1, 2, . . . , n − 2, at least six times. So, for the number of crossings in D we have: crD ( G ∗ + Dn ) = crD ( G ∗ + Dn−2 ) + crD ( T n−1 ∪ T n ) + crD (K6,n−2 , T n−1 ∪ T n ) + crD ( G ∗ , T n−1 ∪ T n )
7
Symmetry 2020, 12, 135
≥6
n − 2 n − 3 2
2
+n−2+
n − 2 2
+ 6( n − 2) + 3 = 6
n n − 1 2
2
+n+
n 2
.
This contradiction with the assumption (2) conﬁrms that D is antipodalfree. Moreover, if r =  R D  1 and s = SD , the assumption (3) together with cr(K6,n ) = 6 n2 n− imply that, in D, if r = 0, then 2 n
there are at least 2 + 1 subgraphs T j for which the edges of G ∗ are crossed just once by them. More precisely: crD ( G ∗ ) + crD ( G ∗ , K6,n ) ≤ crD ( G ∗ ) + 0r + 1s + 2(n − r − s) < n + that is, s + 2( n − r − s ) < n +
n 2
n 2
;
.
(4)
This enforces that 2r + s ≥ n − n2 + 1, and if r = 0, then s ≥ n − n2 + 1 = n2 + 1. Now, j j for T ∈ R D ∪ SD , we discuss the existence of possible conﬁgurations of subgraphs F = G ∗ ∪ T j in D. Case 1: crD ( G ∗ ) = 0. Without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1a. It is obvious that the set R D is empty; that is, r = 0. Thus, we deal with only the conﬁgurations belonging to the nonempty set M D and we discuss over all cardinalities of the set M D in the following subcases: i.
M D  ≥ 3. We consider two subcases. Let us ﬁrst assume that {Ai , A j , Ak } ⊆ M D with i + 2 ≡ j + 1 ≡ k (mod 5). Without lost of generality, let us consider three different subgraphs T n−2 , T n−1 , T n ∈ SD such that F n−2 , F n−1 and F n have conﬁgurations Ai , A j , and Ak , respectively. Then, crD ( T n−2 ∪ T n−1 ∪ T n , T m ) ≥ 14 holds for any T m ∈ SD with m = n − 2, n − 1, n by summing the values in all columns in the considered three rows of Table 1. Moreover, crD ( G ∗ ∪ T n−2 ∪ T n−1 ∪ T n , T m ) ≥ 6 is fulﬁlling for any subgraph T m ∈ SD by Lemma 1. crD ( T n−2 ∪ T n−1 ∪ T n ) ≥ 13 holds by summing of three corresponding values of Table 1 between the considered conﬁgurations Ai , A j , and Ak , by ﬁxing the subgraph G ∗ ∪ T n−2 ∪ T n−1 ∪ T n , crD ( G ∗ + Dn ) = crD (K6,n−3 ) + crD (K6,n−3 , G ∗ ∪ T n−2 ∪ T n−1 ∪ T n ) + crD ( G ∗ ∪ T n−2 ∪ T n−1 ∪ T n )
≥6
n − 3 n − 4
n − 3 n − 4
+ 6n + 9s − 29 2 2 2 n n − 3 n − 4 n n n − 1 . ≥6 + 6n + 9 + 1 − 29 ≥ 6 +n+ 2 2 2 2 2 2 2
+ 15(s − 3) + 6(n − s) + 13 + 3 = 6
In addition, let us assume that M D = {Ai , A j , Ak } with i + 1 ≡ j (mod 5), j + 1 ≡ k (mod 5), and k + 1 ≡ i (mod 5). Without lost of generality, let us consider two different subgraphs T n−1 , T n ∈ SD such that F n−1 and F n have mentioned conﬁgurations Ai and A j , respectively. Then, crD ( G ∗ ∪ T n−1 ∪ T n , T m ) ≥ 1 + 10 = 11 holds for any T m ∈ SD with m = n − 1, n also, by summing the values in Table 1. Hence, by ﬁxing the subgraph G ∗ ∪ T n−1 ∪ T n , crD ( G ∗ + Dn ) = crD (K6,n−2 ) + crD (K6,n−2 , G ∗ ∪ T n−1 ∪ T n ) + crD ( G ∗ ∪ T n−1 ∪ T n )
≥6
ii.
n − 2 n − 3
n − 2 n − 3
+ 4n + 7s − 16 2 2 2 n − 2 n − 3 n n n − 1 n . ≥6 + 4n + 7 + 1 − 16 ≥ 6 +n+ 2 2 2 2 2 2 2
+ 11(s − 2) + 4(n − s) + 4 + 2 = 6
M D  = 2; that is, M D = {Ai , A j } for some i, j ∈ {1, . . . , 5} with i = j. Without lost of generality, let us consider two different subgraphs T n−1 , T n ∈ SD such that F n−1 and F n have mentioned conﬁgurations Ai and A j , respectively. Then, crD ( G ∗ ∪ T n−1 ∪ T n , T m ) ≥ 1 + 10 = 11 holds for any T m ∈ SD with m = n − 1, n also by Table 1. Thus, by ﬁxing the subgraph G ∗ ∪ T n−1 ∪ T n , we are able to use the same inequalities as in the previous subcase.
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Symmetry 2020, 12, 135
iii.
M D  = 1; that is, M D = {A j } for only one j ∈ {1, . . . , 5}. Without lost of generality, let us assume that T n ∈ SD with the conﬁguration A j ∈ M D of the subgraph F n . As M D = {A j }, we have crD ( G ∗ ∪ T n , T k ) ≥ 1 + 6 = 7 for any T k ∈ SD , k = n provided that rotD (tn ) = rotD (tk ), for more see [13]. Hence, by ﬁxing the subgraph G ∗ ∪ T n , crD ( G ∗ + Dn ) = crD (K6,n−1 ) + crD (K6,n−1 , G ∗ ∪ T n ) + crD ( G ∗ ∪ T n ) n − 1 n − 2 + 7( s − 1) + 3( n − s ) + 1 = 6 + 3n + 4s − 6 2 2 2 2 n − 1 n − 2 n n n − 1 n ≥6 + 3n + 4 +1 −6 ≥ 6 +n+ . 2 2 2 2 2 2
≥6
n − 1 n − 2
Case 2: crD ( G ∗ ) = 1 with crD (C6 ( G ∗ )) = 0. At ﬁrst, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1b. Since the set R D can be nonempty, two possible subcases may occur: i.
Let R D be the nonempty set; that is, there is a subgraph T i ∈ R D . Now, for a T i ∈ R D , the reader can easily see that the subgraph Fi = G ∗ ∪ T i is uniquely represented by rotD (ti ) = (165432), and crD ( T i , T j ) ≥ 6 for any T j ∈ R D with j = i provided that rotD (ti ) = rotD (t j ); for more see [13]. Moreover, it is not difﬁcult to verify by a discussion over all possible drawings D that crD ( G ∗ ∪ T i , T k ) ≥ 5 holds for any subgraph T k ∈ SD , and crD ( G ∗ ∪ T i , T k ) ≥ 4 is also fulﬁlling for any subgraph T k ∈ R D ∪ SD . Thus, by ﬁxing the subgraph G ∗ ∪ T i , crD ( G ∗ + Dn ) ≥ 6
n − 1 n − 2 2
+4n + (2r + s) − 5 ≥ 6 ii.
2
+ 6(r − 1) + 5s + 4(n − r − s) + 1 = 6
n − 1 n − 2 2
2
+ 4n + n −
n 2
+1 −5 ≥ 6
n − 1 n − 2 2
n n − 1 2
2
2
+n+
n 2
.
Let R D be the empty set; that is, each subgraph T j crosses the edges of G ∗ at least once in D. Thus, we deal with the conﬁgurations belonging to the nonempty set N D . Let us consider a subgraph T j ∈ SD with the conﬁguration Bi ∈ N D of F j , where i ∈ {1, 2, 3, 4}. Then, the lowerbounds of number of crossings of two conﬁgurations from N conﬁrm that crD ( G ∗ ∪ T j , T k ) ≥ 1 + 4 = 5 holds for any T k ∈ SD , k = j. Moreover, one can also easily verify over all possible drawings D that crD ( G ∗ ∪ T j , T k ) ≥ 4 is true for any subgraph T k ∈ SD . Hence, by ﬁxing the subgraph G∗ ∪ T j , crD ( G ∗ + Dn ) ≥ 6
+4n + s − 3 ≥ 6
n − 1 n − 2 2
2
n − 1 n − 2 2
2
+ 5( s − 1) + 4( n − s ) + 1 + 1 = 6
+ 4n +
n 2
+1 −3 ≥ 6
n − 1 n − 2 2
n n − 1 2
2
2
+n+
n 2
.
In addition, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1e. It is obvious that the set R D is empty; that is, the set SD cannot be empty. Thus, we deal with the conﬁgurations belonging to the nonempty set O D . Note that the lowerbounds of number of crossings of two conﬁgurations from O were already established in Table 2. Since there is a possibility to ﬁnd a subdrawing of G ∗ ∪ T j ∪ T k , in which crD ( G ∗ ∪ T j , T k ) = 3 with T j ∈ SD and T k ∈ SD , we discuss four following subcases: i.
E4 ∈ O D . Without lost of generality, let us assume that T n ∈ SD with the conﬁguration E4 ∈ O D of F n . Only for this subcase, one can easily verify over all possible drawings D for which crD ( G ∗ ∪ T n , T k ) ≥ 4 is true for any subgraph T k ∈ SD . Thus, by ﬁxing the subgraph G ∗ ∪ T n , crD ( G ∗ + Dn ) ≥ 6
n − 1 n − 2 2
2
+ 5( s − 1) + 4( n − s ) + 1 + 1 = 6 9
n − 1 n − 2 2
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Symmetry 2020, 12, 135
+4n + s − 3 ≥ 6
n − 1 n − 2 2
2
+ 4n +
n n − 1 n +1 −3 ≥ 6 +n+ . 2 2 2 2
n
E4 ∈ O D and E3 ∈ O D . Without lost of generality, let us assume that T n ∈ SD with the conﬁguration E3 ∈ O D of F n . In this subcase, crD ( G ∗ ∪ T n , T k ) ≥ 1 + 5 = 6 holds for any subgraph T k ∈ SD , k = n by the remaining values in the third row of Table 2. Hence, by ﬁxing the subgraph G ∗ ∪ T n ,
ii.
crD ( G ∗ + Dn ) ≥ 6
+3n + 3s − 4 ≥ 6
n − 1 n − 2 2
+ 6( s − 1) + 3( n − s ) + 1 + 1 = 6
2
n − 1 n − 2 2
2
+ 3n + 3
n 2
+1 −4 ≥ 6
n − 1 n − 2 2
n n − 1 2
2
2
+n+
n 2
.
O D = {E1 , E2 }. Without lost of generality, let us consider two different subgraphs T n−1 , T n ∈ SD such that F n−1 and F n have mentioned conﬁgurations E1 and E2 , respectively. Then, crD ( G ∗ ∪ T n−1 ∪ T n , T k ) ≥ 1 + 10 = 11 holds for any T k ∈ SD with k = n − 1, n also by Table 2. Thus, by ﬁxing the subgraph G ∗ ∪ T n−1 ∪ T n ,
iii.
crD ( G ∗ + Dn ) ≥ 6
+4n + 7s − 16 ≥ 6
n − 2 n − 3 2
2
n − 2 n − 3 2
2
+ 11(s − 2) + 4(n − s) + 4 + 2 = 6
+ 4n + 7
n 2
+ 1 − 16 ≥ 6
n − 2 n − 3 2
2
n n − 1 2
2
+n+
n 2
.
O D = {Ei } for only one i ∈ {1, 2}. Without lost of generality, let us assume that T n ∈ SD with the conﬁguration E1 of F n . In this subcase, crD ( G ∗ ∪ T n , T k ) ≥ 1 + 6 = 7 holds for any T k ∈ SD , k = n provided that rotD (tn ) = rotD (tk ). Hence, by ﬁxing the subgraph G ∗ ∪ T n ,
iv.
crD ( G ∗ + Dn ) ≥ 6
+3n + 4s − 6 ≥ 6
n − 1 n − 2 2
2
n − 1 n − 2 2
2
+ 7( s − 1) + 3( n − s ) + 1 = 6
+ 3n + 4
n 2
+1 −6 ≥ 6
n − 1 n − 2 2
2
n n − 1 2
2
+n+
n 2
.
Case 3: crD ( G ∗ ) = 2 with crD (C6 ( G ∗ )) = 0. At ﬁrst, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1c. It is obvious that the set R D is empty, that is, the set SD cannot be empty. Our aim is to list again all possible rotations rotD (t j ) which can appear in D if a subgraph T j ∈ SD . Since there is only one subdrawing of F j \ {v1 } represented by the rotation (26543), there are three ways to obtain the subdrawing of F j depending on which edge of G ∗ is crossed by the edge t j v1 . These three possible ways under our consideration can be denoted by Ck , for k = 1, 2, 3. Based on the aforementioned arguments, we assume the drawings shown in Figure 7.
Figure 7. Drawings of three possible conﬁgurations of the subgraph
10
Fj.
Symmetry 2020, 12, 135
Thus the conﬁgurations C1 , C2 , and C3 are represented by the cyclic permutations (132654), (143265), and (165432), respectively. Further, due to the properties of the cyclic rotations we can easily verify that cr(Ci , C j ) ≥ 4 for any i, j ∈ {1, 2, 3}. Moreover, one can also easily verify over all possible drawings D that crD ( G ∗ ∪ T j , T k ) ≥ 4 holds for any subgraph T k ∈ SD , where T j ∈ SD with some conﬁguration Ci of F j . As there is a T j ∈ SD , by ﬁxing the subgraph G ∗ ∪ T j , crD ( G ∗ + Dn ) ≥ 6
+4n + s − 2 ≥ 6
n − 1 n − 2 2
2
n − 1 n − 2 2
2
+ 5( s − 1) + 4( n − s ) + 2 + 1 = 6
+ 4n +
n 2
+1 −2 ≥ 6
n − 1 n − 2 2
n n − 1 2
2
2
+n+
n 2
.
In addition, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1d. In this case, by applying the same process, we obtain two possible forms of rotation rotD (t j ) for T j ∈ SD . Namely, the rotations (165423) and (165432) if the edge t j v2 crosses either the edge v3 v4 or the edge v3 v5 of G ∗ , respectively. Further, they satisfy also the same properties like in the previous subcase, i.e., the same lower bounds of numbers of crossings on the edges of the subgraph G ∗ ∪ T j by any T k , k = j. Hence, we are able to use the same ﬁxing of the subgraph G ∗ ∪ T j for obtaining a contradiction with the number of crossings in D. Finally, without loss of generality, we can consider the drawing of G ∗ with the vertex notation like that in Figure 1f. In this case, the set R D is empty; that is, the set SD cannot be empty. Thus, we can deal with the conﬁgurations belonging to the nonempty set P D . Recall that the lowerbounds of number of crossings of two conﬁgurations from P were already established in Table 3. Further, we can apply the same idea and also the same arguments as for the conﬁgurations Ei ∈ O D , with i = 1, 2, 3, in the subcases ii.–iv. of Case 2. Case 4: crD ( G ∗ ) ≥ 1 with crD (C6 ( G ∗ )) ≥ 1. For all possible subdrawings of the graph G ∗ with at least one crossing among edges of C6 ( G ∗ ), and also with the possibility of obtaining a subgraph T j that crosses the edges of G ∗ at most once, one of the ideas of the previous subcases can be applied. We have shown, in all cases, that there is no good drawing D of the graph G ∗ + Dn with fewer 1 than 6 n2 n− + n + n2 crossings. This completes the proof of the main theorem. 2 4. Conclusions Determining the crossing number of a graph G + Dn is an essential step in establishing the so far unknown values of the numbers of crossings of graphs G + Pn and G + Cn , where Pn and Cn are the path and the cycle on n vertices, respectively. Using the result in Theorem 1 and the optimal drawing of G ∗ + Dn in Figure 6, we are able to postulate that cr( G ∗ + Pn ) and cr( G ∗ + Cn ) are at least one more 1 than cr( G ∗ + Dn ) = 6 n2 n− + n + n2 . 2 Funding: This research received no external funding. Acknowledgments: This work was supported by the internal faculty research project number FEI201739. Conﬂicts of Interest: The author declares no conﬂict of interest.
References 1. 2. 3. 4. 5.
Garey, M.R.; Johnson, D.S. Crossing number is NPcomplete. SIAM J. Algebraic. Discrete Methods 1983, 4, 312–316. Klešˇc, M. The join of graphs and crossing numbers. Electron. Notes Discret. Math. 2007, 28, 349–355. Kulli, V.R.; Muddebihal, M.H. Characterization of join graphs with crossing number zero. Far East J. Appl. Math. 2001, 5, 87–97. Kleitman, D.J. The crossing number of K5,n . J. Comb. Theory 1970, 9, 315–323. Klešˇc, M.; Schrötter, Š. The crossing numbers of join products of paths with graphs of order four. Discuss. Math. Graph Theory 2011, 31, 312–331.
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6. 7. 8. 9. 10. 11. 12.
13. 14.
Berežný, Š.; Staš, M. Cyclic permutations and crossing numbers of join products of symmetric graph of order six. Carpathian J. Math. 2018, 34, 143–155. Klešˇc, M. The crossing numbers of join of the special graph on six vertices with path and cycle. Discret. Math. 2010, 310, 1475–1481. Staš, M. Cyclic permutations: Crossing numbers of the join products of graphs. In Proceedings of the Aplimat 2018: 17th Conference on Applied Mathematics, Bratislava, Slovakia, 6–8 February 2018; pp. 979–987. Staš, M. Determining crossing numbers of graphs of order six using cyclic permutations. Bull. Aust. Math. Soc. 2018, 98, 353–362. HernándezVélez, C.; Medina, C.; Salazar G. The optimal drawing of K5,n . Electron. J. Comb. 2014, 21, 29. Berežný, Š.; Buša J., Jr.; Staš, M. Software solution of the algorithm of the cyclicorder graph. Acta Electrotech. Inform. 2018, 18, 3–10. Klešˇc, M.; Schrötter, Š. The crossing numbers of join of paths and cycles with two graphs of order five. In Lecture Notes in Computer Science: Mathematical Modeling and Computational Science; Springer: Berlin/Heidelberg, Germany, 2012; Volume 7125, pp. 160–167. Woodall, D.R. Cyclicorder graphs and Zarankiewicz’s crossing number conjecture. J. Graph Theory 1993, 17, 657–671. Chimani, M.; Wiedera, T. An ILPbased proof system for the crossing number problem. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA 2016), Aarhus, Denmark, 22–24 August 2016; Volume 29, pp. 1–13. c 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
12
SS symmetry Article
Bounds for the Generalized Distance Eigenvalues of a Graph Abdollah Alhevaz 1, *, Maryam Baghipur 1 , Hilal Ahmad Ganie 2 and Yilun Shang 3 1 2 3
*
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood P.O. Box: 3163619995161, Iran; [email protected] Department of Mathematics, University of Kashmir, Srinagar 190006, India; [email protected] Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK; [email protected] Correspondence: [email protected] or [email protected]
Received: 1 October 2019; Accepted: 13 December 2019; Published: 17 December 2019
Abstract: Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and Tr ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by Dα ( G ) the generalized distance matrix, i.e., Dα ( G ) = αTr ( G ) + (1 − α) D ( G ), where α ∈ [0, 1]. In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α, improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the kth generalized distance eigenvalue. Keywords: distance matrix (spectrum); distance signlees Laplacian matrix (spectrum); (generalized) distance matrix; spectral radius; transmission regular graph MSC: Primary: 05C50, 05C12; Secondary: 15A18
1. Introduction We will consider simple ﬁnite graphs in this paper. A (simple) graph is denoted by G = (V ( G ), E( G )), where V ( G ) = {v1 , v2 , . . . , vn } represents its vertex set and E( G ) represents its edge set. The order of G is the number of vertices represented by n = V ( G ) and its size is the number of edges represented by m =  E( G ). The neighborhood N (v) of a vertex v consists of the set of vertices that are adjacent to it. The degree dG (v) or simply d(v) is the number of vertices in N (v). In a regular graph, all its vertices have the same degree. Let duv be the distance between two vertices u, v ∈ V ( G ). It is deﬁned as the length of a shortest path. D ( G ) = (duv )u,v∈V (G) is called the distance matrix of G. G is the complement of the graph G. It has the same vertex set with G but its edge set consists of the edges not present in G. Moreover, the complete graph Kn , the complete bipartite graph Ks,t , the path Pn , and the cycle Cn are deﬁned in the conventional way. The transmission TrG (v) of a vertex v is the sum of the distances from v to all other vertices in G, i.e., TrG (v) = ∑ duv . A graph G is said to be ktransmission regular if TrG (v) = k, for each v ∈ V ( G ). u ∈V ( G )
The transmission (also called the Wiener index) of a graph G, denoted by W ( G ), is the sum of distances between all unordered pairs of vertices in G. We have W ( G ) = 12 ∑ TrG (v). v ∈V ( G )
Symmetry 2019, 11, 1529; doi:10.3390/sym11121529
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Symmetry 2019, 11, 1529
For a vertex vi ∈ V ( G ), TrG (vi ) is also referred to as the transmission degree, or shortly Tri . The sequence of transmission degrees { Tr1 , Tr2 , . . . , Trn } is the transmission degree sequence of the graph. Ti =
n
∑ dij Tr j is called the second transmission degree of vi .
j =1
Distance matrix and its spectrum has been studied extensively in the literature, see e.g., [6]. Compared to adjacency matrix, distance matrix encapsulates more information such as a wide range of walkrelated parameters, which can be applicable in thermodynamic calculations and have some biological applications in terms of molecular characterization. It is known that embedding theory and molecular stability have to do with graph distance matrix. Almost all results obtained for the distance matrix of trees were extended to the case of weighted trees by Bapat [12] and Bapat et al. [13]. Not only different classes of graphs but the deﬁnition of distance matrix has been extended. Indeed, Bapat et al. [14] generalized the concept of the distance matrix to that of qanalogue of the distance matrix. Let Tr ( G ) = diag( Tr1 , Tr2 , . . . , Trn ) be the diagonal matrix of vertex transmissions of G. The works [7–9] introduced the distance Laplacian and the distance signless Laplacian matrix for a connected graph G. The matrix D L ( G ) = Tr ( G ) − D ( G ) is referred to as the distance Laplacian matrix of G, while the matrix D Q ( G ) = Tr ( G ) + D ( G ) is the distance signless Laplacian matrix of G. Spectral properties of D ( G ) and D Q ( G ) have been extensively studied since then. Let A be the adjacency matrix and Deg( G ) = diag(d1 , d2 , . . . , dn ) be the degree matrix G. Q( G ) = Deg( G ) + A is the signless Laplacian matrix of G. This matrix has been put forth by Cvetkovic in [16] and since then studied extensively by many researchers. For detailed coverage of this research see [17–20] and the references therein. To digging out the contribution of these summands in Q( G ), Nikiforov in [33] proposed to study the αadjacency matrix Aα ( G ) of a graph G given by Aα ( G ) = α Deg( G ) + (1 − α) A, where α ∈ [0, 1]. We see that Aα ( G ) is a convex combination of the matrices A and Deg( G ). Since A0 ( G ) = A and 2A1/2 ( G ) = Q( G ), the matrix Aα ( G ) can underpin a uniﬁed theory of A and Q( G ). Motivated by [33], Cui et al. [15] introduced the convex combinations Dα ( G ) of Tr ( G ) and D ( G ). The matrix Dα ( G ) = αTr ( G ) + (1 − α) D ( G ), 0 ≤ α ≤ 1, is called generalized distance matrix of G. Therefore the generalized distance matrix can be applied to the study of other less general constructions. This not only gives new results for several matrices simultaneously, but also serves the uniﬁcation of known theorems. Since the matrix Dα ( G ) is real and symmetric, its eigenvalues can be arranged as: ∂1 ≥ ∂2 ≥ · · · ≥ ∂n , where ∂1 is referred to as the generalized distance spectral radius of G. For simplicity, ∂( G ) is the shorthand for ∂1 ( G ). By the PerronFrobenius theorem, ∂( G ) is unique and it has a unique generalized distance Perron vector, X, which is positive. This is due to the fact that Dα ( G ) is nonnegative and irreducible. A column vector X = ( x1 , x2 , . . . , xn ) T ∈ Rn is a function deﬁned on V ( G ). We have X (vi ) = xi for all i. Moreover, n
X T Dα ( G ) X = α ∑ Tr (vi ) xi2 + 2(1 − α) i =1
∑
1≤ i < j ≤ n
d ( vi , v j ) xi x j ,
and λ has an eigenvector X if and only if X = 0 and n
λxv = αTr (vi ) xi + (1 − α) ∑ d(vi , v j ) x j . j =1
They are often referred to as the (λ, x )eigenequations of G. If X ∈ Rn has at least one nonnegative element and it is normalized, then in the light of the Rayleigh’s principle, it can be seen that ∂( G ) ≥ X T Dα ( G ) X,
14
Symmetry 2019, 11, 1529
where the equality holds if and only if X becomes the generalized distance Perron vector of G. Spectral graph theory has been an active research ﬁeld for the past decades, in which for example distance signless Laplacian spectrum has been intensively explored. The work [41] identiﬁed the graphs with minimum distance signless Laplacian spectral radius among some special classes of graphs. The unique graphs with minimum and secondminimum distance signless Laplacian spectral radii among all bicyclic graphs of the same order are identiﬁed in [40]. In [24], the authors show some bounding inequalities for distance signless Laplacian spectral radius by utilizing vertex transmissions. In [26], chromatic number is used to derive a lower bound for distance signless Laplacian spectral radius. The distance signless Laplacian spectrum has varies connections with other interesting graph topics such as chromatic number [10]; domination and independence numbers [21], Estrada indices [4,5,22,23,34–36,38], cospectrality [11,42], multiplicity of the distance (signless) Laplacian eigenvalues [25,29,30] and many more, see e.g., [1–3,27,28,32]. The rest of the paper is organized as follows. In Section 2, we obtain some bounds for the generalized distance spectral radius of graphs using the diameter, the order, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α. We then characterize the extremal graphs. In Section 3, we are devoted to derive new upper and lower bounds for the kth generalized distance eigenvalue of the graph G using signless Laplacian eigenvalues and the αadjacency eigenvalues. 2. Bounds on Generalized Distance Spectral Radius In this section, we obtain bounds for the generalized distance spectral radius, in terms of the diameter, the order, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α. The following lemma can be found in [31]. Lemma 1. If A is an n × n nonnegative matrix with the spectral radius λ( A) and row sums r1 , r2 , . . . , rn , then min ri ≤ λ( A) ≤ max ri .
1≤ i ≤ n
1≤ i ≤ n
Moreover, if A is irreducible, then both of the equalities holds if and only if the row sums of A are all equal. The following gives an upper bound for ∂( G ), in terms of the order n, the diameter d and the minimum degree δ of the graph G. Theorem 1. Let G be a connected graph of order n having diameter d and minimum degree δ. Then ∂( G ) ≤ dn −
d ( d − 1) − 1 − δ ( d − 1), 2
(1)
with equality if and only if G is a regular graph with diameter ≤ 2. Proof. First, it is easily seen that, Tr p =
n
∑ d jp
j =1
≤ d p + 2 + 3 + · · · + (d − 1) + d(n − 1 − d p − (d − 2)) = dn −
d ( d − 1) − 1 − d p ( d − 1), 2
15
for all p = 1, 2, . . . , n.
(2)
Symmetry 2019, 11, 1529
Let Trmax = max{ TrG (vi ) : 1 ≤ i ≤ n}. For a matrix A denote λ( A) its largest eigenvalue. We have ∂( G ) = λ α( Tr ( G )) + (1 − α) D ( G ) ≤ αλ Tr ( G ) + (1 − α)λ D ( G )
≤ αTrmax + (1 − α) Trmax = Trmax . Applying Equation (2), the inequality follows. Suppose that G is regular graph with diameter less than or equal to two, then all coordinates of the generalized distance Perron vector of G are equal. If d = 1, then G ∼ = Kn and ∂ = n − 1. Thus equality in (1) holds. If d = 2, we get ∂( G ) = di + 2(n − 1 − di ) = 2n − 2 − di , and the equality in (1) holds. Note that the equality in (1) holds if and only if all coordinates the generalized distance Perron vector are equal, and hence Dα ( G ) has equal row sums. Conversely, suppose that equality in (1) holds. This will force inequalities above to become equations. Then we get Tr1 = Tr2 = · · · = Trn = Trmax , hence all the transmissions of the vertices are equal and so G is a transmission regular graph. If d ≥ 3, then from the above argument, for every vertex vi , there is exactly one vertex v j with dG (vi , v j ) = 2, and thus d = 3, and for a vertex vs of eccentricity 2, ∂ ( G ) x s = d s x s + 2( n − 1 − d s ) x s =
3n −
3(3 − 1) − 1 − d s (3 − 1) x s , 2
implying that ds = n − 2, giving that G = P4 . But the Dα ( P4 ) is not transmission regular graph. Therefore, G turns out to be regular and its diameter can not be greater than 2. Taking α = 12 in Theorem 1, we immediately get the following bound for the distance signless Laplacian spectral radius ρ1Q ( G ), which was proved recently in [27]. Corollary 1. ([27], Theorem 2.6) Let G be a connected graph of order n ≥ 3, with minimum degree δ1 , second minimum degree δ2 and diameter d. Then ρ1Q ( G ) ≤ 2dn − d(d − 1) − 2 − (δ1 + δ2 )(d − 1), with equality if and only if G is (transmission) regular graph of diameter d ≤ 2. Proof. As 2D 1 ( G ) = D Q ( G ), letting δ = δ1 in Theorem 1, we have 2
ρ1Q ( G ) = 2∂( G ) ≤ 2dn − d(d − 1) − 2 − 2δ1 (d − 1) ≤ 2dn − d(d − 1) − 2 − (δ1 + δ2 )(d − 1), and the result follows. Next, the generalized distance spectral radius ∂( G ) of a connected graph and its complement is characterized in terms of a NordhausGaddum type inequality. Corollary 2. Let G be a graph of order n, such that both G and its complement G are connected. Let δ and Δ be the minimum degree and the maximum degree of G, respectively. Then ∂( G ) + ∂( G ) ≤ 2nk − (t − 1)(t + n + δ − Δ − 1) − 2, where k = max{d, d}, t = min{d, d} and d, d are the diameters of G and G, respectively.
16
Symmetry 2019, 11, 1529
Proof. Let δ denote the minimum degree of G. Then δ = n − 1 − Δ, and by Theorem 1, we have ∂( G ) + ∂( G )
¯ ¯ d ( d − 1) ¯ − d(d − 1) − 1 − δ¯(d¯ − 1) − 1 − δ(d − 1) + dn 2 2 1 ¯ ¯ ¯ n(d + d) − (d(d − 1) + d(d − 1)) − 2 − δ(d − 1) − (n − 1 − Δ)(d¯ − 1) 2 2nk − (t − 1)(t + n + δ − Δ − 1) − 2.
≤ dn − = ≤
The following gives an upper bound for ∂( G ), in terms of the order n, the minimum degree δ = δ1 and the second minimum degree δ2 of the graph G. Theorem 2. Let G be a connected graph of order n having minimum degree δ1 and second minimum degree δ2 . Then for s = δ1 + δ2 , we have ∂( G ) ≤ where Θ = dn −
αΨ +
α2 Ψ2 + 4(1 − 2α)Θ , 2
(3)
− 1 − δ1 (d − 1) dn − d(d2−1) − 1 − δ2 (d − 1) and Ψ = 2dn − d(d − 1) − 2 − s(d − 1). Also equality holds if and only if G is a regular graph with diameter at most two. d ( d −1) 2
Proof. Let X = ( x1 , x2 , . . . , xn ) T be the generalized distance Perron vector of graph G and let xi = max{ xk k = 1, 2, . . . , n} and x j = maxk =i { xk k = 1, 2, . . . , n}. From the ith equation of Dα ( G ) X = ∂( G ) X, we obtain ∂xi = αTri xi + (1 − α)
n
∑
k =1,k =i
dik xk ≤ αTri xi + (1 − α) Tri x j .
(4)
Similarly, from the jth equation of Dα ( G ) X = ∂( G ) X, we obtain ∂x j = αTr j x j + (1 − α)
n
∑
k =1,k = j
d jk xk ≤ αTr j x j + (1 − α) Tr j xi .
(5)
Now, by (2), we have,
d ( d − 1) d ( d − 1) ∂ − α dn − xi ≤ (1 − α) dn − − 1 − d i ( d − 1) − 1 − d i ( d − 1) x j 2 2 d ( d − 1) d ( d − 1) ∂ − α dn − x j ≤ (1 − α) dn − − 1 − d j ( d − 1) − 1 − d j ( d − 1) x i . 2 2 Multiplying the corresponding sides of these inequalities and using the fact that xk > 0 for all k, we obtain ∂2 − α(2dn − d(d − 1) − 2 − (d − 1)(di + d j ))∂ − (1 − 2α)ξ i ξ j ≤ 0, where ξ l = dn − ∂( G ) ≤
d ( d −1) 2
− 1 − dl (d − 1), l = i, j, which in turn gives
α(2dn − d(d − 1) − 2 − s(d − 1)) +
α2 (2dn − d(d − 1) − 2 − s(d − 1))2 + 4(1 − 2α)Θ . 2
Now, using di + d j ≥ δ1 + δ2 , the result follows. Suppose that equality occurs in (3), then equality occurs in each of the above inequalities. If equality occurs in (4) and (5), the we obtain xi = xk , for all k = 1, 2, . . . , n giving that G is a 17
Symmetry 2019, 11, 1529
transmission regular graph. Also, equality in (2), similar to that of Theorem 1, gives that G is a graph of diameter at most two and equality in di + d j ≥ δ1 + δ2 gives that G is a regular graph. Combining all these it follows that equality occurs in (3) if G is a regular graph of diameter at most two. Conversely, if G is a connected δregular graph of diameter at most two, then ∂( G ) = Tri = d ( d −1) dn − 2 − 1 − di (d − 1). Also α(2dn − d(d − 1) − 2 − s(d − 1)) +
= =
α2 (2dn − d(d − 1) − 2 − s(d − 1))2 + 4(1 − 2α)Θ 2 α(2dn − d(d − 1) − 2 − s(d − 1)) + (2dn − d(d − 1) − 2 − s(d − 1))(1 − α) 2 d ( d − 1) dn − − 1 − δ ( d − 1) = ∂ ( G ). 2
That completes the proof. Remark 1. For any connected graph G of order n having minimum degree δ, the upper bound given by Theorem 2 is better than the upper bound given by Theorem 1. As α(2dn − d(d − 1) − 2 − s(d − 1)) +
≤ = =
α2 (2dn − d(d − 1) − 2 − s(d − 1))2 + 4(1 − 2α)Θ , 2 α(2dn − d(d − 1) − 2 − 2δ(d − 1)) + α2 (2dn − d(d − 1) − 2 − 2δ(d − 1))2 + 4(1 − 2α)Φ , 2 α(2dn − d(d − 1) − 2 − 2δ(d − 1)) + (2dn − d(d − 1) − 2 − 2δ(d − 1))(1 − α) 2 d ( d − 1) dn − − 1 − δ ( d − 1), 2
where Φ = (2dn − d(d − 1) − 2 − 2δ(d − 1))2 . The following gives an upper bound for ∂( G ) by using quantities like transmission degrees as well as second transmission degrees. Theorem 3. If the transmission degree sequence and the second transmission degree sequence of G are { Tr1 , Tr2 , . . . , Trn } and { T1 , T2 , . . . , Tn }, respectively, then ∂( G ) ≤ max
⎫ ⎧ ⎨ − β + β2 + 4(αTr2 + (1 − α) Ti + βTri ) ⎬ i
1≤ i ≤ n ⎩
⎭
2
,
(6)
where β ≥ 0 is an unknown parameter. Equality occurs if and only if G is a transmission regular graph. Proof. Let X = ( x1 , . . . , xn ) be the generalized distance Perron vector of G and xi = max{ x j  j = 1, 2, . . . , n}. Since ∂ ( G )2 X
= ( Dα ( G ))2 X = (αTr + (1 − α) D )2 X = α2 Tr2 X + α(1 − α) TrDX + α(1 − α) DTrX + (1 − α)2 D2 X,
we have ∂2 ( G ) xi = α2 Tri2 xi + α(1 − α) Tri
n
n
j =1
j =1
n
n
∑ dij x j + α(1 − α) ∑ dij Tr j x j + (1 − α)2 ∑ ∑ dij d jk xk .
18
j =1 k =1
Symmetry 2019, 11, 1529
Now, we consider a simple quadratic function of ∂( G ) :
(∂2 ( G ) + β∂( G )) X
= (α2 Tr2 X + α(1 − α) TrDX + α(1 − α) DTrX + (1 − α)2 D2 X ) +
β(αTrX + (1 − α) DX ).
Considering the ith equation, we have
(∂2 ( G ) + β∂( G )) xi
= α2 Tri2 xi + α(1 − α) Tri + (1 − α )2
n
n
∑∑
n
n
∑ dij x j + α(1 − α) ∑ dij Tr j x j
j =1
n
αTri xi + α(1 − α) ∑ dij x j
dij d jk xk + β
j =1 k =1
j =1
.
j =1
It is easy to see that the inequalities below are true α(1 − α) Tri
(1 − α )2
n
∑ dij x j ≤ α(1 − α)Tri2 xi ,
j =1
n
n
α(1 − α) ∑ dij Tr j x j ≤ α(1 − α) Ti xi ,
n
∑ ∑ d jk dij xk ≤ (1 − α)2 Ti xi ,
j =1 k =1
j =1
n
(1 − α) ∑ dij x j ≤ (1 − α) Tri xi . j =1
Hence, we have
(∂2 ( G ) + β∂( G )) xi ≤ αTri2 xi − αTi xi + Ti xi + βTri xi ⇒ ∂2 ( G ) + β∂( G ) − (αTri2 − (α − 1) Ti + βTri ) ≤ 0 − β + β2 + 4(αTri2 − (α − 1) Ti + βTri ) ⇒ ∂( G ) ≤ . 2 From this the result follows. Now, suppose that equality occurs in (6), then each of the above inequalities in the above argument occur as equalities. Since each of the inequalities α(1 − α) Tri
n
∑ dij x j ≤ α(1 − α)Tri2 xi ,
j =1
n
α(1 − α) ∑ dij Tr j x j ≤ α(1 − α) Ti xi j =1
and
(1 − α )2
n
n
∑ ∑ d jk dij xk ≤ (1 − α)2 Ti xi ,
j =1 k =1
n
(1 − α) ∑ dij x j ≤ (1 − α) Tri xi , j =1
occur as equalities if and only if G is a transmission regular graph. It follows that equality occurs in (6) if and only if G is a transmission regular graph. That completes the proof. The following upper bound for the generalized distance spectral radius ∂( G ) was obtained in [15]: ∂( G ) ≤ max
1≤ i ≤ n
αTri2 + (1 − α) Ti
with equality if and only if αTri2 + (1 − α) Ti is same for i.
19
,
(7)
Symmetry 2019, 11, 1529
Remark 2. For a connected graph G having transmission degree sequence { Tr1 , Tr2 , . . . , Trn } and the second transmission degree sequence { T1 , T2 , . . . , Tn }, provided that Ti ≤ Tri2 for all i, we have
−β +
β2 + 4αTri2 + 4(1 − α) Ti + 4βTri 2
≤
αTri2 + (1 − α) Ti .
Therefore, the upper bound given by Theorem 3 is better than the upper bound given by (7). If, in particular we take the parameter β in Theorem 3 equal to the vertex covering number τ, the edge covering number, the clique number ω, the independence number, the domination number, the generalized distance rank, minimum transmission degree, maximum transmission degree, etc., then Theorem 3 gives an upper bound for ∂( G ), in terms of the vertex covering number τ, the edge covering number, the clique number ω, the independence number, the domination number, the generalized distance rank, minimum transmission degree, maximum transmission degree, etc. Let xi = min{ x j  j = 1, 2, . . . , n} be the minimum among the entries of the generalized distance Perron vector X = ( x1 , . . . , xn ) of the graph G. Proceeding similar to Theorem 3, we obtain the following lower bound for ∂( G ), in terms of the transmission degrees, the second transmission degrees and a parameter β. Theorem 4. If the transmission degree sequence and the second transmission degree sequence of G are { Tr1 , Tr2 , . . . , Trn } and { T1 , T2 , . . . , Tn }, respectively, then ∂( G ) ≥ min
⎧ ⎫ ⎨ − β + β2 + 4(αTr2 + (1 − α) Ti + βTri ) ⎬ i
1≤ i ≤ n ⎩
⎭
2
,
where β ≥ 0 is an unknown parameter. Equality occurs if and only if G is a transmission regular graph. Proof. Similar to the proof of Theorem 3 and is omitted. The following lower bound for the generalized distance spectral radius was obtained in [15]: ∂( G ) ≥ min
1≤ i ≤ n
αTri2 + (1 − α) Ti
,
(8)
with equality if and only if αTri2 + (1 − α) Ti is same for i. Similar to Remark 2, it can be seen that the lower bound given by Theorem 4 is better than the lower bound given by (8) for all graphs G with Ti ≥ Tri2 , for all i. Again, if in particular we take the parameter β in Theorem 4 equal to the vertex covering number τ, the edge covering number, the clique number ω, the independence number, the domination number, the generalized distance rank, minimum transmission degree, maximum transmission degree, etc, then Theorem 4 gives a lower bound for ∂( G ), in terms of the vertex covering number τ, the edge covering number, the clique number ω, the independence number, the domination number, the generalized distance rank, minimum transmission degree, maximum transmission degree, etc. G1 ∇ G2 is referred to as join of G1 and G2 . It is deﬁned by joining every vertex in G1 to every vertex in G2 . Example 1. (a) Let C4 be the cycle of order 4. One can easily see that C4 is a 4transmission regular graph and the generalized distance spectrum of C4 is {4, 4α, 6α − 2[2] }. Hence, ∂(C4 ) = 4. Moreover, the transmission degree sequence and the second transmission degree sequence of C4 are {4, 4, 4, 4} and
20
Symmetry 2019, 11, 1529
{16, 16, 16, 16}, respectively. Now, putting β = Trmax = 4 in the given bound of Theorem 3, we can see that the equality holds: ∂(C4 ) ≤
−4 +
√ 16 + 4(16α + 16(1 − α) + 16) −4 + 144 = = 4. 2 2
(b) Let Wn+1 be the wheel graph of order n + 1. It is well known that Wn+1 = Cn ∇K1 . The distance signless Laplacian matrix of W5 is ⎛
5 ⎜1 ⎜ ⎜ D Q (W5 ) = ⎜2 ⎜ ⎝1 1
1 5 1 2 1
2 1 5 1 1
⎞ 1 1⎟ ⎟ ⎟ 1⎟ . ⎟ 1⎠ 4
1 2 1 5 1
Hence the distance signless Laplacian spectrum of W5 is spec(W5 ) = √
#
$ √ √ 13+ 41 13− 41 5 3 [2] , , 2, 2 , and then 4 4
the distance signless Laplacian spectral radius is ρ1Q (W5 ) = 13+4 41 . Also, the transmission degree sequence and the second transmission degree sequence of W5 are {5, 5, 5, 5, 4} and {24, 24, 24, 24, 20}, respectively. As D 1 ( G ) = 12 D Q ( G ), taking α = 12 and β = Trmax = 5 in the given bound of Theorem 3, we immediately 2
get the following upper bound for the distance signless Laplacian spectral radius ρ1Q (W5 ):
−5 + 1 Q ρ (W5 ) ≤ 2 1
√
√ −5 + 223 25 + 50 + 48 + 100 , = 2 2
which implies that ρ1Q (W5 ) ≤ −5 +
√
223 9.93.
3. Bounds for the kth Generalized Distance Eigenvalue In this section, we discuss the relationship between the generalized distance eigenvalues and the other graph parameters. The following lemma can be found in [37]. Lemma 2. Let X and Y be Hermitian matrices of order n such that Z = X + Y, and denote the eigenvalues of a matrix M by λ1 ≥ λ2 ≥ · · · ≥ λn .Then λk ( Z ) ≤ λ j ( X ) + λk− j+1 (Y ), n ≥ k ≥ j ≥ 1, λk ( Z ) ≥ λ j ( X ) + λk− j+n (Y ), n ≥ j ≥ k ≥ 1, where λi ( M ) is the ith largest eigenvalue of the matrix M. Any equality above holds if and only if a unit vector can be an eigenvector corresponding to each of the three eigenvalues. The following gives a relation between the generalized distance eigenvalues of the graph G of diameter 2 and the signless Laplacain eigenvalues of the complement G of the graph G. It also gives a relation between generalized distance eigenvalues of the graph G of diameter greater than or equal to 3 with the αadjacency eigenvalues of the complement G of the graph G. Theorem 5. Let G be a connected graph of order n ≥ 4 having diameter d. Let G be the complement of G and let q1 ≥ q2 ≥ · · · ≥ qn be the signless Laplacian eigenvalues of G. If d = 2, then for all k = 1, 2, . . . , n, we have
(3α − 1)n − 2α + (1 − 2α)dk + (1 − α)qk ≤ ∂k ( G ) ≤ (2n − 2)α + (1 − 2α)dk + (1 − α)qk .
21
Symmetry 2019, 11, 1529
Equality occurs on the right if and only if k = 1 and G is a transmission regular graph and on the left if and only if k = 1 and G is a transmission regular graph. If d ≥ 3, then for all k = 1, 2, . . . , n, we have αn − 1 + λk ( Aα ( G )) + λn ( M ) ≤ ∂k ( G ) ≤ n − 1 + λk ( Aα ( G )) + λ1 ( M ), where Aα ( G ) = α Deg( G ) + (1 − α) A is the αadjacency matrix of G and M = αTr ( G ) + (1 − α) M with M = (mij ) a symmetric matrix of order n having mij = max{0, dij − 2}, dij is the distance between the vertices vi , v j and Tr ( G ) = diag( Tr1 , Tr2 , . . . , Trn ), Tri = ∑ (dij − 2). dij ≥3
Proof. Let G be a connected graph of order n ≥ 4 having diameter d. Let Deg( G ) = diag(n − 1 − d1 , n − 1 − d2 , . . . , n − 1 − dn ) be the diagonal matrix of vertex degrees of G. Suppose that diameter d of G is two, then transmission degree Tri = 2n − 2 − di , for all i, then the distance matrix of G can be written as D ( G ) = A + 2A, where A and A are the adjacency matrices of G and G, respectively. We have Dα ( G ) = αTr ( G ) + (1 − α) D ( G ) = α(2n − 2) I − α Deg( G ) + (1 − α)( A + 2A)
= α(2n − 2) I − α Deg( G ) + (1 − α)( A + A) + (1 − α) A = (3nα − n − 2α) I + (1 − α) J + (1 − 2α) Deg( G ) + (1 − α) Q( G ), where I is the identity matrix and J is the all one matrix of order n. Taking Y = (3nα − n − 2α) I + (1 − 2α) Deg( G ) + (1 − α) Q( G ), X = (1 − α) J, j = 1 in the ﬁrst inequality of Lemma 2 and using the fact that spec( J ) = {n, 0[n−1] }, it follows that ∂k ( G ) ≤ (2n − 2)α + (1 − 2α)dk + (1 − α)qk ,
for all
k = 1, 2, . . . , n.
(9)
Again, taking Y = (3nα − n − 2α) I + (1 − 2α) Deg( G ) + (1 − α) Q( G ), X = (1 − α) J and j = n in the second inequality of Lemma 2, it follows that ∂k ( G ) ≥ (3α − 1)n − 2α + (1 − 2α)dk + (1 − α)qk ,
for all
k = 1, 2, . . . , n.
(10)
Combining (9) and (10) the ﬁrst inequality follows. Equality occurs in ﬁrst inequality if and only if equality occurs in (9) and (10). Suppose that equality occurs in (9), then by Lemma 2, the eigenvalues ∂k , (3n − 2)α − n + (1 − 2α)dk + (1 − α)qk and n(1 − α) of the matrices Dα ( G ), X and Y have the same unit eigenvector. Since 1 = n1 (1, 1, . . . , 1) T is the unit eigenvector of Y for the eigenvalue n(1 − α), it follows that equality occurs in (9) if and only if 1 is the unit eigenvector for each of the matrices Dα ( G ), X and Y. This gives that G is a transmission regular graph and G is a regular graph. Since a graph of diameter 2 is regular if and only if it is transmission regular and complement of a regular graph is regular. Using the fact that for a connected graph G the unit vector 1 is an eigenvector for the eigenvalue ∂1 if and only if G is transmission regular graph, it follows that equality occurs in ﬁrst inequality if and only if k = 1 and G is a transmission regular graph. Suppose that equality occurs in (10), then again by Lemma 2, the eigenvalues ∂k , (3n − 2)α − n + (1 − 2α)dk + (1 − α)qk and 0 of the matrices Dα ( G ), X and Y have the same unit eigenvector x. Since Jx = 0, it follows that x T 1 = 0. Using the fact that the matrix J is symmetric(so its normalized eigenvectors are orthogonal [43]), we conclude that the vector 1 belongs to the set of eigenvectors of the matrix J and so of the matrices Dα ( G ), X. Now, 1 is an eigenvector of the matrices Dα ( G ) and X, gives that G is a regular graph. Since for a regular graph of diameter 2 any eigenvector of Q( G ) and Dα ( G ) is orthogonal to 1, it follows that equality occurs in (10) if and only if k = 1 and G is a regular graph. If d ≥ 3, we deﬁne the matrix M = (mij ) of order n, where mij = max{0, dij − 2}, dij is the distance
22
Symmetry 2019, 11, 1529
between the vertices vi and v j . The transmission of a vertex vi can be written as Tri = di + 2di + Tri , where Tri = ∑ (dij − 2), is the contribution from the vertices which are at distance more than two dij ≥3
from vi . For Tr ( G ) = diag( Tr1 , Tr2 , . . . , Trn ), we have Dα ( G ) = αTr ( G ) + (1 − α) D ( G ) = α Deg( G ) + 2α Deg( G ) + αTr ( G ) + (1 − α)( A + 2A + M )
= α(Deg( G ) + Deg( G )) + (1 − α)( A + A) + (α Deg( G ) + (1 − α) A) + (αTr ( G ) + (1 − α) M) = Dα ( K n ) + A α ( G ) + M , where Aα ( G ) is the αadjacency matrix of G and M = αTr ( G ) + (1 − α) M. Taking X = Dα (Kn ), Y = Aα ( G ) + M and j = 1 in the ﬁrst inequality of Lemma 2 and using the fact that spec( Dα (Kn )) = {n − 1, αn − 1[n−1] }, it follows that ∂ k ( G ) ≤ n − 1 + λ k ( A α ( G ) + M ),
for all
k = 1, 2, . . . , n.
Again, taking Y = Aα ( G ), X = M and j = 1 in the ﬁrst inequality of Lemma 2, we obtain ∂k ( G ) ≤ n − 1 + λk ( Aα ( G )) + λ1 ( M ),
for all
k = 1, 2, . . . , n.
(11)
Similarly, taking X = Dα (Kn ), Y = Aα ( G ) + M and j = n and then Y = Aα ( G ), X = M and j = n in the second inequality of Lemma 2, we obtain ∂k ( G ) ≥ αn − 1 + λk ( Aα ( G )) + λn ( M ),
for all
k = 1, 2, . . . , n.
(12)
From (11) and (12) the second inequality follows. That completes the proof. It can be seen that the matrix M deﬁned in Theorem 5 is positive semideﬁnite for all Therefore, we have the following observation from Theorem 5. Corollary 3. Let G be a connected graph of order n ≥ 4 having diameter d ≥ 3. If ∂k ( G ) ≥ αn − 1 + λk ( Aα ( G )),
for all
1 2
1 2
≤ α ≤ 1.
≤ α ≤ 1, then
k = 1, 2, . . . , n,
where Aα ( G ) = α Deg( G ) + (1 − α) A is the αadjacency matrix of G. It is clear from Corollary 3 that for 12 ≤ α ≤ 1, any lower bound for the αadjacency λk ( Aα ( G )) gives a lower bound for ∂k and conversely any upper bound for ∂ gives an upper bound for λk ( Aα ( G )). We note that Theorem 5 generalizes one of the Theorems (namely Theorem 3.8) given in [8]. Example 2. (a) Let Cn be a cycle of order n. It is well known (see [7]) that Cn is a ktransmission regular 2 2 graph with k = n4 if n is even and k = n 4−1 if n is odd. Let n = 4. It is clear that the distance spectrum of the graph C4 is {4, 0, −2[2] }. Also, since C4 is a 4transmission regular graph, then Tr (C4 ) = 4I4 and so Dα (C4 ) = 4αI4 + (1 − α) D (C4 ). Hence the generalized distance spectrum of C4 is {4, 4α, 6α − 2[2] }. Moreover, the signless Laplacian spectrum of C4 is {2[2] , 0[2] }. Since the diameter of C4 is 2, hence, applying Theorem 5, for k = 1, we have, 4α = 4(3α − 1) − 2α + 2(1 − 2α) + 2(1 − α) ≤ ∂1 (C4 ) = 4 ≤ 6α + 2(1 − 2α) + 2(1 − α) = 4, which shows that the equality occurs on right for k = 1 and transmission regular graph C4 . Also, for k = 2, we have 4α = 4(3α − 1) − 2α + 2(1 − 2α) + 2(1 − α) ≤ ∂2 (C4 ) = 4α ≤ 6α + 2(1 − 2α) + 2(1 − α) = 4,
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Symmetry 2019, 11, 1529
which shows that the equality occurs on left for k = 2 and transmission regular graph C4 . (b) Let C6 be a cycle of order 6. It is clear that the distance spectrum of the graph C6 is {9, 0[2] , −1, −4[2] }. Since C6 is a 9transmission regular graph, then Tr (C6 ) = 9I6 and so Dα (C6 ) = 9αI6 + (1 − α) D (C6 ). Hence, the generalized distance spectrum of C6 is {9, 9α[2] , 10α − 1, 13α − 4[2] }. Also, the αadjacency spectrum of C6 is {3, 2α + 1, 3α[2] , 5α − 2[2] }. Let M be the matrix deﬁned by the Theorem 5, hence the spectrum of M is {1[3] , 2α − 1[3] }. Since diameter of the graph C6 is 3, hence, applying Theorem 5, for k = 1, we have 8α + 1 = 6α − 1 + 3 + 2α − 1 ≤ ∂1 (C6 ) = 9 ≤ 5 + 3 + 1 = 9. Also for k = 2, we have 10α − 1 = 6α − 1 + 2α + 1 + 2α − 1 ≤ ∂2 (C6 ) = 9α ≤ 5 + 2α + 1 + 1 = 2α + 7. We need the following lemma proved by Hoffman and Wielandt [39]. Lemma 3. Suppose we have C = A + B. Here, all these matrices are symmetric and have order n. Suppose they have the eigenvalues αi , β i , and γi , where 1 ≤ i ≤ n, respectively arranged in nonincreasing order. Therefore, ∑in=1 (γi − αi )2 ≤ ∑in=1 β2i . The following gives relation between generalized distance spectrum and distance spectrum for a simple connected graph G. We use [n] to denote the set of {1, 2, . . . , n}. For each subset S of [n], we use Sc to denote [n] − S. Theorem 6. Let G be a connected graph of order n and let μ1 , . . . , μn be the eigenvalues of the distance matrix of G. Then for each nonempty subset S = {r1 , r2 , . . . , rk } of [n], we have the following inequalities: 2kαW ( G ) −
≤ ≤
k (n − k ) n ∑in=1 α2 Tri2 − 4α2 W 2 ( G ) n
∑ ( ∂ i + ( α − 1) μ i )
i∈S
2kαW ( G ) +
k (n − k ) n ∑in=1 α2 Tri2 − 4α2 W 2 ( G ) n
.
Proof. Since Dα ( G ) = αTr ( G ) + (1 − α) D ( G ), then by the fact that 2αW ( G ) = ∑in=1 (∂i + (α − 1)μi ), we get 2αW ( G ) − ∑i∈S (∂i + (α − 1)μi ) = ∑i∈SC (∂i + (α − 1)μi ). By CauchySchwarz inequality, we further have that
2 2αW ( G ) − ∑ (∂i + (α − 1)μi ) i∈S
≤
∑
12
i ∈ SC
∑ ( ∂ i + ( α − 1) μ i )2 .
i ∈ SC
Therefore 2
2αW ( G ) − ∑ (∂i + (α − 1)μi )
≤ (n − k)
i∈S
n
∑ ( ∂ i + ( α − 1) μ i )2 − ∑ ( ∂ i + ( α − 1) μ i )2 .
i =1
i∈S
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Symmetry 2019, 11, 1529
By Lemma 3, we have that 2
2αW ( G ) − ∑ (∂i + (α − 1)μi )
+ ( n − k ) ∑ ( ∂ i + ( α − 1) μ i )2
i∈S
n
i∈S
n
≤ (n − k) ∑ (∂i + (α − 1)μi ) ≤ (n − k) ∑ α2 Tri2 . 2
i =1
i =1
Again by CauchySchwarz inequality, we have that 2 % 2 n−k n−k ( ∂ i + ( α − 1) μ i ) ∑ ( ∂ i + ( α − 1) μ i ) = ∑ k k i∈S i∈S n−k ∑ k ∑ ( ∂ i + ( α − 1) μ i )2 = ( n − k ) ∑ ( ∂ i + ( α − 1) μ i )2 . i∈S i∈S i∈S
≤
Therefore, we have the following inequality 2
2αW ( G ) − ∑ (∂i + (α − 1)μi )
+
i∈S
n−k k
2
∑ ( ∂ i + ( α − 1) μ i )
i∈S
n
≤ (n − k) ∑ α2 Tri2 . i =1
Solving the quadratic inequality for ∑i∈S (∂i + (α − 1)μi ), so we complete the proof. Notice that ∑in=1 (∂i − αTri ) = 0 and by Lemma 3, we also have ∑in=1 (∂i − αTri )2 ≤ (1 − α)2 ∑in=1 μ2i = 2(1 − α)2 ∑1n≤i< j≤n d2ij . We can similarly prove the following theorem. Theorem 7. Let G be a connected graph of order n. Then for each nonempty subset S = {r1 , r2 , . . . , rk } of [n], we have: & ' & & & 2k (n − k )(1 − α)2 ∑1≤i< j≤n d2ij & & . & ∑ (∂i − αTri )& ≤ & &i ∈S n We conclude by giving the following bounds for the kth largest generalized distance eigenvalue of a graph. Theorem 8. Assume G is connected and is of order n. Suppose it has diameter d and δ is its minimum degree. Let ϕ( G )
2 2 α n ( n − 1) = min n2 (n − 1) + (1 − α)2 d2 − 4α2 W 2 ( G ), 4 2 d ( d − 1 ) 2 2 2 n α nd − − 1 − δ(d − 1) + (1 − α) n(n − 1)d − 4α2 W 2 ( G ) . 2
Then for k = 1, . . . , n, 1 n
(
% 2αW ( G ) −
k−1 ϕ( G ) n−k+1
)
1 ≤ ∂k ( G ) ≤ n
25
(
% 2αW ( G ) +
n−k ϕ( G ) k
) .
(13)
Symmetry 2019, 11, 1529
Proof. First we prove the upper bound. It is clear that trace( Dα2 ( G )) =
k
n
i =1
i = k +1
∑ ∂2i + ∑
∂2i ≥
(∑ik=1 ∂i )2 (∑in=k+1 ∂i )2 + . k n−k
Let Mk = ∑ik=1 ∂i . Then Mk2 (2αW ( G ) − Mk )2 , + k n−k
trace( Dα2 ( G )) ≥ which implies M 1 ∂k ( G ) ≤ k ≤ k n
(
% 2αW ( G ) +
n−k [n · trace( Dα2 ( G )) − 4α2 W 2 ( G )] k
) .
We observe that n · trace( Dα2 ( G )) − 4α2 W 2 ( G )
n
= nα2 ∑ Tri2 + 2n(1 − α)2 i =1
∑
1≤ i < j ≤ n
(dij )2 − 4α2 W 2 ( G )
n3 ( n − 1)2 n ( n − 1) 2 + 2n(1 − α)2 d − 4α2 W 2 ( G ) 4 2 2 2 α n ( n − 1) n2 ( n − 1) + (1 − α)2 d2 − 4α2 W 2 ( G ), 4
≤ nα2 = since Tri ≤
n ( n −1) , 2
and
n · trace( Dα2 ( G )) − 4α2 W 2 ( G ) n
= nα2 ∑ Tri2 + 2n(1 − α)2 i =1
∑
1≤ i < j ≤ n
(dij )2 − 4α2 W 2 ( G )
2 d ( d − 1) n ( n − 1) 2 ≤ nα2 nd − − 1 − δ(d − 1) + 2n(1 − α)2 d − 4α2 W 2 ( G ) 2 2 2 d ( d − 1) − 1 − δ(d − 1) + (1 − α)2 n(n − 1)d2 − 4α2 W 2 ( G ), = n α2 nd − 2 d ( d −1)
since Tri ≤ nd − 2 − 1 − di (d − 1). Hence, we get the righthand side of the inequality (13). Now, we prove the lower bound. Let Nk = ∑in=k ∂i . Then we have trace( Dα2 ( G ))
=
k −1
∑
i =1
∂2i
n
+∑
i=k
∂2i
≥ =
∑ik=−11 ∂i k−1
2
+
∑in=k ∂i
2
n−k+1
Nk2 (2αW ( G ) − Nk )2 + . k−1 n−k+1
Hence ∂k ( G ) ≥
Nk 1 ≥ n−k+1 n
(
% 2αW ( G ) −
k−1 [n · trace( Dα2 ( G )) − 4α2 W 2 ( G )] n−k+1
) ,
and we get the lefthand side of the inequality (13). By a chemical tree, we mean a tree which has all vertices of degree less than or equal to 4.
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Symmetry 2019, 11, 1529
Example 3. In Figure 1, we depicted a chemical tree of order n = 5.
Figure 1. A chemical tree T.
The distance matrix of T is
⎛
0 ⎜1 ⎜ ⎜ D ( T ) = ⎜2 ⎜ ⎝3 3
1 0 1 2 2
2 1 0 1 1
3 2 1 0 2
⎞ 3 2⎟ ⎟ ⎟ 1⎟ . ⎟ 2⎠ 0
Let μ1 , . . . , μ5 be the distance eigenvalues of the tree T. Then one can easily see that μ1 = 7.46, μ2 = −0.51, μ3 = −1.08, μ4 = −2 and μ5 = −3.86. Note that, as D0 ( T ) = D ( T ), taking α = 0 in Theorem 8, then for n = 5 we get −6
k −1 6− k
≤ μk ≤ 6
5− k k ,
for any 1 ≤ k ≤ 5. For example, −6 ≤ μ1 ≤ 12 and −3 ≤ μ2 ≤ 7.3.
4. Conclusions Motivated by an article entitled “Merging the A and Qspectral theories” by V. Nikiforov [33], recently, Cui et al. [15] dealt with the integration of spectra of distance matrix and distance signless Laplacian through elegant convex combinations accommodating vertex transmissions as well as distance matrix. For α ∈ [0, 1], the generalized distance matrix is known as Dα ( G ) = αTr ( G ) + (1 − α) D ( G ). Our results shed light on some properties of Dα ( G ) and contribute to establishing new inequalities (such as lower and upper bounds) connecting varied interesting graph invariants. We established some bounds for the generalized distance spectral radius for a connected graph using various identities like the number of vertices n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α, improving some bounds recently given in the literature. We also characterized the extremal graphs attaining these bounds. Notice that the current work mainly focuses to determine some bounds for the spectral radius (largest eigenvalue) of the generalized distance matrix. It would be interesting to derive some bounds for other important eigenvalues such as the smallest eigenvalue as well as the second largest eigenvalue of this matrix. Author Contributions: conceptualization, A.A., M.B. and H.A.G.; formal analysis, A.A., M.B., H.A.G. and Y.S.; writing—original draft preparation, A.A., M.B. and H.A.G.; writing—review and editing, A.A., M.B., H.A.G. and Y.S.; project administration, A.A.; funding acquisition, Y.S. Funding: Y. Shang was supported by UoA Flexible Fund No. 201920A1001 from Northumbria University. Acknowledgments: The authors would like to thank the academic editor and the four anonymous referees for their constructive comments that helped improve the quality of the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.
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Lu, L.; Huang, Q.; Huang, X. On graphs whose smallest distance (signless Laplacian) eigenvalue has large multiplicity. Linear Multilinear Algebra 2018, 66, 2218–2231. [CrossRef] Min´c, H. Nonnegative Matrices; John Wiley & Sons: New York, NY, USA, 1988. Lin, H.; Zhou, B. The distance spectral radius of trees. Linear Multilinear Algebra 2019, 67, 370–390. [CrossRef] Nikiforov, V. Merging the A and Qspectral theories. Appl. Anal. Discrete Math. 2017, 11, 81–107. [CrossRef] Shang, Y. Distance Estrada index of random graphs. Linear Multilinear Algebra 2015, 63, 466–471. [CrossRef] Shang, Y. Estimating the distance Estrada index. Kuwait J. Sci. 2016, 43, 14–19. Shang, Y. Bounds of distance Estrada index of graphs. Ars Combin. 2016, 128, 287–294. So, W. Commutativity and spectra of Hermitian matrices. Linear Algebra Appl. 1994, 212–213, 121–129. [CrossRef] Yang, J.; You, L.; Gutman, I. Bounds on the distance Laplacian energy of graphs. Kragujevac J. Math. 2013, 37, 245–255. Wilkinson, J.H. The Algebraic Eigenvalue Problem; Oxford University Press: New York, NY, USA, 1965. Xing, R.; Zhou, B. On the distance and distance signless Laplacian spectral radii of bicyclic graphs. Linear Algebra Appl. 2013, 439, 3955–3963. [CrossRef] Xing, R.; Zhou, B.; Li, J. On the distance signless Laplacian spectral radius of graphs. Linear Multilinear Algebra 2014, 62, 1377–1387. [CrossRef] Xue, J.; Liu, S.; Shu, J. The complements of path and cycle are determined by their distance (signless) Laplacian spectra. Appl. Math. Comput. 2018, 328, 137–143. [CrossRef] Zhang, F. Matrix Theory: Basic Results and Techniques; Springer: New York, NY, USA, 1999. c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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SS symmetry Article
Weak Embeddable HypernearRings Jelena Daki´c 1 , Sanja Janˇci´cRašovi´c 1 and Irina Cristea 2, * 1 2
*
Department of Mathematics, Faculty of Natural Science and Mathematics, University of Montenegro, 81000 Podgorica, Montenegro Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia Correspondence: [email protected] or [email protected]; Tel.: +386053315395
Received: 28 June 2019; Accepted: 18 July 2019; Published: 1 August 2019
Abstract: In this paper we extend one of the main problems of nearrings to the framework of algebraic hypercompositional structures. This problem states that every nearring is isomorphic with a nearring of the transformations of a group. First we endow the set of all multitransformations of a hypergroup (not necessarily abelian) with a general hypernearring structure, called the multitransformation general hypernearring associated with a hypergroup. Then we show that any hypernearring can be weakly embedded into a multitransformation general hypernearring, generalizing the similar classical theorem on nearrings. Several properties of hypernearrings related with this property are discussed and illustrated also by examples. Keywords: hypernearring; multitransformation; embedding
1. Introduction Generally speaking, the embedding of an algebraic structure into another one requires the existence of an injective map between the two algebraic objects, that also preserves the structure, i.e., a monomorphism. The most natural, canonical and wellknown embeddings are those of numbers: the natural numbers into integers, the integers into the rational numbers, the rational numbers into the real numbers and the real numbers into the complex numbers. One important type of rings is that one of the endomorphisms of an abelian group under function pointwise addition and composition of functions. It is well known that every ring is isomorphic with a subring of such a ring of endomorphisms. But this result holds only in the commutative case, since the set of the endomorphisms of a nonabelian group is no longer closed under addition. This aspect motivates the interest in studying nearrings, that appear to have applications also in characterizing transformations of a group. More exactly, the set of all transformations of a group G, i.e., T ( G ) = { f : G → G } can be endowed with a nearring structure under pointwise addition and composition of mappings, such a nearring being called the transformation nearring of the group G. In 1959 Berman and Silverman [1] claimed that every nearring is isomorphic with a nearring of transformations. At that time only some hints were presented, while a direct and clear proof of this result appeared in Malone and Heatherly [2] almost ten years later. Since T ( G ) has an identity, it immediately follows that any nearring can be embedded in a nearring with identity. Moreover, in the same paper [2], it was proved that a group ( H, +) can be embedded in a group ( G, +) if and only if the nearring T0 ( H ), consisting of all transformations of H which multiplicatively commute with the zero transformation, can be embedded into the similar nearring T0 ( G ) on G under a kernelpreserving monomorphism of nearrings. Similarly to nearrings, but in the framework of algebraic hyperstructures, Daši´c [3] deﬁned the hypernearrings as hyperstructures with the additive part being a quasicanonical hypergroup [4,5] (called also a polygroup [6,7]), and the multiplicative part being a semigroup with a bilaterally
Symmetry 2019, 11, 964; doi:10.3390/sym11080964
30
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Symmetry 2019, 11, 964
absorbing element, such that the multiplication is distributive with respect to the hyperaddition on the lefthand side. Later on, this algebraic hyperstructure was called a strongly distributive hypernearring, or a zerosymmetric hypernearring, while in a hypernearring the distributivity property was replaced by the “inclusive distributivity” from the left (or right) side. Moreover, when the additive part is a hypergroup and all the other properties related to the multiplication are conserved, we talk about a general hypernearring [8]. The distributivity property is important also in other types of hyperstructures, see e.g., [9]. A detailed discussion about the terminology related to hypernearrings is included in [10]. In the same paper, the authors deﬁned on the set of all transformations of a quasicanonical hypergroup that preserves the zero element a hyperaddition and a multiplication (as the composition of functions) in such a way to obtain a hypernearring. More general, the set of all transformations of a hypergroup (not necessarily commutative) together with the same hyperaddition and multiplication is a strongly distributive hypernearring [3]. In this note we will extend the study to the set of all multimappings (or multitransformations) of a (nonabelian) hypergroup, deﬁning ﬁrst a structure of (left) general hypernearring, called the multitransformation general hypernearring associated with a hypergroup. Then we will show that any hypernearring can be weakly embedded into a multitransformation general hypernearring, generalizing the similar classical theorem on nearrings [2]. Besides, under same conditions, any additive hypernearring is weakly embeddable into the additive hypernearring of the transformations of a hypergroup with identity element that commute multiplicatively with the zerofunction. The paper ends with some conclusive ideas and suggestions of future works on this topic. 2. Preliminaries We start with some basic deﬁnitions and results in the framework of hypernearrings and nearrings of group mappings. For further properties of these concepts we refer the reader to the papers [2,3,11,12] and the fundamental books [13–15]. For the consistence of our study, regarding hypernearrings we keep the terminology established and explained in [8,16]. First we recall the deﬁnition introduced by Daši´c in 1978. Deﬁnition 1. [12] A hypernearring is an algebraic system ( R, +, ·), where R is a nonempty set endowed with a hyperoperation + : R × R → P∗ ( R) and an operation · : R × R → R, satisfying the following three axioms: 1.
( R, +) is a quasicanonical hypergroup (named also polygroup [6]), meaning that: (a) (b) (c) (d)
2. 3.
x + (y + z) = ( x + y) + z for any x, y, z ∈ R, there exists 0 ∈ R such that, for any x ∈ R, x + 0 = 0 + x = { x }, for any x ∈ R there exists a unique element − x ∈ R, such that 0 ∈ x + (− x ) ∩ (− x ) + x, for any x, y, z ∈ R, z ∈ x + y implies that x ∈ z + (−y), y ∈ (− x ) + z.
( R, ·) is a semigroup endowed with a twosided absorbing element 0, i.e., for any x ∈ R, x · 0 = 0 · x = 0. The operation “·” is distributive with respect to the hyperoperation “+” from the lefthand side: for any x, y, z ∈ R, there is x · (y + z) = x · y + x · z.
This kind of hypernearring was called by Gontineac [11] a zerosymmetric hypernearring. In our previous works [10,16], regarding the distributivity, we kept the Vougiouklis’ terminology [17], and therefore, we say that a hypernearring is a hyperstructure ( R, +, ·) satisfying the above mentioned axioms 1. and 2., and the new one: 3 . The operation “·” is inclusively distributive with respect to the hyperoperation “+” from the lefthand side: for any x, y, z ∈ R, x · (y + z) ⊆ x · y + x · z. Accordingly, the Daši´c ’s hypernearring (satisfying the axioms 1., 2., and 3.) is called strongly distributive hypernearring. Furthermore, if the additive part is a hypergroup (and not a polygroup), then we talk about a more general type of hypernearrings.
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Symmetry 2019, 11, 964
Deﬁnition 2. [8] A general (left) hypernearring is an algebraic structure ( R, +, ·) such that ( R, +) is a hypergroup, ( R, ·) is a semihypergroup and the hyperoperation “·” is inclusively distributive with respect to the hyperoperation “+” from the lefthand side, i.e., x · (y + z) ⊆ x · y + x · z, for any x, y, z ∈ R. If in the third condition the equality is valid, then the structure ( R, +, ·) is called strongly distributive general (left) hypernearring. Besides, if the multiplicative part ( R, ·) is only a semigroup (instead of a semihypergroup), we get the notion of general (left) additive hypernearring. Deﬁnition 3. Let ( R1 , +, ·) and ( R2 , +, ·) be two general hypernearrings. A map ρ : R1 → R2 is called an inclusion homomorphism if the following conditions are satisﬁed: 1. 2.
ρ( x + y) ⊆ ρ( x ) + ρ(y) ρ( x · y) ⊆ ρ( x ) · ρ(y) for all x, y ∈ R1 .
A map ρ is called a good (strong) homomorphism if in the conditions 1. and 2. the equality is valid. In the second part of this section we will brieﬂy recall the fundamentals on nearrings of group mappings. A left nearring ( N, +, ·) is a nonempty set endowed with two binary operations, the addition + and the multiplication ·, such that ( N, +) is a group (not necessarily abelian) with the neutral element 0, ( N, ·) is a semigroup, and the multiplication is distributive with respect to the addition from the lefthand side. Similarly, we have a right nearring. Several examples of nearrings are obtained on the set of “nonlinear” mappings and here we will see two of them. Let ( G, +) be a group (not necessarily commutative) and let T ( G ) be the set of all functions from G to G. On T ( G ) deﬁne two binary operations: “+” is the pointwise addition of functions, while the multiplication “·” is the composition of functions. Then ( T ( G ), +, ·) is a (left) nearring, called the transformation nearring on the group G. Moreover, let T0 ( G ) be the subnearring of T ( G ) consisting of the functions of T ( G ) that commute multiplicatively with the zero function, i.e., T0 ( G ) = { f ∈ T ( G )  f (0) = 0}. These two nearrings, T ( G ) and T0 ( G ), have a fundamental role in embeddings. Already in 1959, it was claimed by Berman and Silverman [1] that every nearring is isomorphic with a nearring of transformations. One year later the proof was given by the same authors, but using an elaborate terminology and methodology. Here below we recall this result together with other related properties, as presented by Malone and Heatherly [2]. Theorem 1. [2] Let ( R, +, ·) be a nearring. If ( G, +) is any group containing ( R, +) as a proper subgroup, then ( R, +, ·) can be embedded in the transformation nearring T ( G ). Corollary 1. [2] Every nearring can be embedded in a nearring with identity. Theorem 2. [2] A group ( H, +) can be embedded in a group ( G, +) if and only if T0 ( H ) can be embedded in T0 ( G ) by a nearring monomorphism which is kernelpreserving. Theorem 3. [2] A group ( H, +) can be embedded in a group ( G, +) if and only if the nearring T ( H ) can be embedded in the nearring T ( G ). 3. Weak Embeddable HypernearRings In this section we aim to extend the results related to embeddings of nearrings to the case of hypernearrings. In this respect, instead of a group ( G, +) we will consider a hypergroup ( H, +) and then the set of all multimappings on H, which we endow with a structure of general hypernearring. Theorem 4. Let ( H, +) be a hypergroup (not necessarily abelian) and T ∗ ( H ) = {h : H → P∗ ( H )} the set of all multimappings of the hypergroup ( H, +). Deﬁne, for all ( f , g) ∈ T ∗ ( H ) × T ∗ ( H ), the following hyperoperations: f ⊕ g = {h ∈ T ∗ ( H )  (∀ x ∈ H ) h( x ) ⊆ f ( x ) + g( x )} 32
Symmetry 2019, 11, 964
f g = {h ∈ T ∗ ( H )  (∀ x ∈ H ) h( x ) ⊆ g( f ( x )) =
g(u)}.
u∈ f ( x )
The structure ( T ∗ ( H ), ⊕, ) is a (left) general hypernearring. Proof. For any f , g ∈ T ∗ ( H ) it holds: f ⊕ g = ∅. Indeed, for any x ∈ H, it holds f ( x ) = ∅ and g( x ) = ∅ and thus, f ( x ) + g( x ) = ∅. Therefore, for the map h : H → P∗ ( H ) deﬁned by: h( x ) = f ( x ) + g( x ) for all x ∈ H, it holds h ∈ f ⊕ g. Now, we prove that the hyperoperation ⊕ is associative. Let f , g, h ∈ T ∗ ( H ) and set L = ( f ⊕ g) ⊕ h =
=
{ h ⊕ h  h ∈ f ⊕ g} =
{h ⊕ h  (∀ x ∈ H ) h ( x ) ⊆ f ( x ) + g( x )}.
Thus, if h ∈ L, then, for all x ∈ H, it holds: h ( x ) ⊆ h ( x ) + h( x ) ⊆ ( f ( x ) + g( x )) + h( x ). Conversely, if h is an element of T ∗ ( H ) such that: h ( x ) ⊆ ( f ( x ) + g( x )) + h( x ), for all x ∈ H, and if we choose h such that h ( x ) = f ( x ) + g( x ) for all x ∈ H, then h ∈ f ⊕ g and h ∈ h ⊕ h i.e., h ∈ L. So, L = {h ∈ T ∗ ( H )(∀ x ∈ H )h ( x ) ⊆ ( f ( x ) + g( x )) + h( x )}. On the other side, take D = f ⊕ ( g ⊕ h). Then, D = { h ∈ T ∗ ( H )(∀ x ∈ H )h ( x ) ⊆ f ( x ) + ( g( x ) + h( x ))}. By the associativity of the hyperoperation + we obtain that L = D, meaning that the hyperoperation ⊕ is associative. Let f , g ∈ T ∗ ( H ). We prove that the equation f ∈ g ⊕ a has a solution a ∈ T ∗ ( H ). If we set a( x ) = H, for all x ∈ H, then a ∈ T ∗ ( H ) and for all x ∈ H it holds g( x ) + a( x ) = H ⊇ f ( x ). So, f ∈ g ⊕ a. Similarly, the equation f ∈ a ⊕ g has a solution in T ∗ ( H ). Thus, ( T ∗ ( H ), ⊕) is a hypergroup. Now, we show that ( T ∗ ( H ), ) is a semihypergroup. Let f , g ∈ T ∗ ( H ). For all x ∈ H it holds g( x ) = ∅ and so g( f ( x )) = ∅. Let h : H → P∗ ( H ) be a multimapping deﬁned by h( x ) = g( f ( x )), for all x ∈ H. Obviously, h ∈ f g and so f g = ∅. Let us prove that is a associative. Let f , g, h ∈ T ∗ ( H ). Set: L = ( f g) h =
{h h  h ∈ f g} = {h h  (∀ x ∈ H ) h ( x ) ⊆ g( f ( x ))} =
= {h  (∀ x ∈ H ) h ( x ) ⊆ h(h ( x )) ∧ h ( x ) ⊆ g( f ( x ))}. So, if h ∈ L, then h ( x ) ⊆ h( g( f ( x ))), for all x ∈ H. On the other side, if h ∈ T ∗ ( H ) and h ( x ) ⊆ h( g( f ( x ))) for all x ∈ H, then we choose h ∈ T ∗ ( H ) such that h ( x ) = g( f ( x )) and consequently we obtain that h ⊆ h(h ( x )). Thus, h ∈ L. So, L = { h ∈ T ∗ ( H )  (∀ x ∈ H ) h ( x ) ⊆ h( g( f ( x )))}. Similarly, D = f ( g h) = {h  (∀ x ∈ H ) h ( x ) ⊆ h( g( f ( x )))}. Thus, L = D. It remains to prove that the hyperoperation ⊕ is inclusively distributive with respect to the * hyperoperation on the lefthand side. Let f , g, h ∈ T ∗ ( H ). Set L = f ( g ⊕ h) = { f h  h ∈ * ∗ g ⊕ h} = { f h  h ∈ T ( H ) ∧ (∀ x )h ( x ) ⊆ g( x ) + h( x )}. So, if k ∈ L then for all x ∈ H it holds: k( x ) ⊆ h ( f ( x )) ⊆ g( f ( x )) + h( f ( x )). * On the other hand, D = ( f g) ⊕ ( f h) = {k1 ⊕ k2 k1 ∈ f g, k2 ∈ f h}. Let k ∈ L. Choose, ∗ k1 , k2 ∈ T ( H ) such that k1 ( x ) = g( f ( x )) and k2 ( x ) = h( f ( x )) for all x ∈ H. Then k1 ∈ f g and k2 ∈ f h. Thus, k ( x ) ⊆ k1 ( x ) + k2 ( x ) for all x ∈ H, i.e., k ∈ k1 ⊕ k2 and k1 ∈ f g, k2 ∈ f h. So, k ∈ D. Therefore, L ⊆ D. Deﬁnition 4. T ∗ ( H ) is called the multitransformations general hypernearring on the hypergroup H. Remark 1. Let ( G, +) be a group and T ( G ) be the transformations nearring on G. Obviously, T ( G ) ⊂ T ∗ ( G ) = { f : G → P∗ ( G )} and, for all f , g ∈ T ( G ), it holds: f ⊕ g = f + g, f g = f · g, meaning that the hyperoperations deﬁned in Theorem 4 are the same as the operations in Theorem 1. It follows that T ( G ) is a sub(hyper)nearring of ( T ∗ ( G ), ⊕, ).
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Symmetry 2019, 11, 964
Deﬁnition 5. We say that the hypernearring ( R1 , +, ·) is weak embeddable (by short W − embeddable) in the hypernearring ( R2 , +, ·) if there exists an injective inclusion homomorphism μ : R1 → R2 . The next theorem is a generalization of Theorem 1 [5]. Theorem 5. For every general hypernearring ( R, +, ·) there exists a hypergroup ( H, +) such that R is W − embeddable in the associated hypernearring T ∗ ( H ). Proof. Let ( R, +, ·) be a hypernearring and let ( H, +) be a hypergroup such that ( R, +) is a proper subhypergroup of ( H, +). For a ﬁxed element r ∈ R we deﬁne a multimapping f r : H → P∗ ( H ) as follows ( g · r, if g ∈ R f r ( g) = r, if g ∈ H \ R. Let us deﬁne now the mapping μ : R → T ∗ ( H ) as μ(r ) = f r , which is an inclusion homomorphism. Indeed, if a, b ∈ R then we have μ( a + b) = { f c  c ∈ a + b} and μ( a) ⊕ μ(b) = f a ⊕ f b = {h  (∀ g ∈ H ) h( g) ⊆ f a ( g) + f b ( g)}. Consider c ∈ a + b and g ∈ H. If g ∈ R, then f c ( g) = g · c ⊆ g · ( a + b) ⊆ g · a + g · b = f a ( g) + f b ( g). If g ∈ H \ R, then f c ( g) = c ∈ a + b = f a ( g) + f b ( g). It follows that, for all g ∈ H, we have f c ( g) ⊆ f a ( g) + f b ( g) and therefore f c ∈ μ( a) ⊕ μ(b), meaning that μ( a + b) ⊆ μ( a) ⊕ μ(b). Similarly, there is μ( a · b) = { f c  c ∈ a · b} and μ( a) μ(b) = f a f b = {h ∈ T ∗ ( H )  (∀ g ∈ H ) h( g) ⊆ f b ( f a ( g))}. Let c ∈ a · b. Then, for g ∈ R, it holds: f c ( g) = g · c ⊆ g · ( a · b) = ( g · a) · b = f b ( f a ( g)). If g ∈ H \ R, then there is f c ( g) = c ∈ a · b = f b ( a) = f b ( f a ( g)). Thus, f c ∈ μ( a) μ(b) and so μ( a b) ⊆ μ( a) μ(b). Based on Deﬁnition 3, we conclude that μ is an inclusive homomorphism. It remains to show that μ is injective. If μ( a) = μ(b), then for all g ∈ H, it holds f a ( g) = f b ( g). So, if we choose g ∈ H \ R, then we get that a = f a ( g) = f b ( g) = b. These all show that the general hypernearring R is Wembeddable in T ∗ ( H ). Remark 2. If ( R, +, ·) is a nearring such that ( R, +) is a proper subgroup of a group ( G, +), then for a ﬁxed r ∈ R the multimapping f r constructed in the proof of Theorem 5 is in fact a map from G to G, since in this case the multiplication · is an ordinary operation, i.e., g · r ∈ G, for all g ∈ R. Thus f r : G → G and thereby μ( R) ⊆ T ( G ). By consequence μ : R → T ( G ) is an ordinary monomorphism. In other words, Theorem 5 is a generalization of Theorem 1. Example 1. Let ( R, +, ·) be a left nearring. Let P1 and P2 be nonempty subsets of R such that R · P1 ⊆ P1 and P1 ⊆ Z ( R), where Z ( R) is the center of R, i.e., Z ( R) = { x ∈ R  (∀y ∈ R) x + y = y + x }. For any ( x, y) ∈ R2 deﬁne: x ⊕ P1 y = x + y + P1 , x P2 y = xP2 y. Then the structure ( R, ⊕ P1 , P2 ) is a general left hypernearring [8,18]. Let H = R ∪ { a} and deﬁne on H the hyperoperation ⊕P as follows: ( x ⊕P1 y =
x ⊕ P1 y, if x, y ∈ R H, if x = a ∨ y = a.
It is clear that H is a hypergroup such that ( R, +) is a proper subhypergroup of ( H, +). Besides, based on Theorem 5, for every r ∈ R the multimapping f r : H → P∗ ( H ) is deﬁned as ( f r ( g) =
g P2 r, r,
if g ∈ R = if g = a
34
(
gP2 r, r,
if g ∈ R if g = a.
Symmetry 2019, 11, 964
Clearly it follows that μ : R → P∗ ( H ), deﬁned by μ(r ) = f r , is an inclusive homomorphism, so the general left hypernearring ( R, ⊕ P1 , P2 ) is Wembeddable in T ∗ ( H ). Example 2. Consider the semigroup (N, ·) of natural numbers with the standard multiplication operation and the order “≤”. Deﬁne on it the hyperoperations +≤ and ·≤ as follows: x +≤ y = {z  x ≤ z ∨ y ≤ z} x · ≤ y = { z  x · y ≤ z }. Then the structure (N, +≤ , ·≤ ) is a strongly distributive general hypernearring (in fact it is a hyperring). This follows from Theorem 4.3 [19]. Furthermore, for any a ∈ / N, it can be easily veriﬁed that (N, +≤ ) is a proper subhypergroup of (N ∪ { a}, +≤ ), where the hyperoperation +≤ is deﬁned by: ( x +≤
y=
x +≤ y, if x, y ∈ N N ∪ { a}, if x = a ∨ y = a.
In this case, for a ﬁxed n ∈ N, we can deﬁne the multimapping f n : N ∪ { a} → P∗ (N ∪ { a}) as follows: ( f n ( g) =
(
g ·≤ n, if g ∈ N = n, if g = a
{k ∈ N  g · n ≤ k}, if g ∈ N n, if g = a
and therefore the mapping μ : N → P∗ (N ∪ { a}) is an inclusive homomorphism. Again this shows that the general hypernearring (N, +≤ , ·≤ ) is Wembeddable in T ∗ (N ∪ { a}). Example 3. Let R = {0, 1, 2, 3}. Consider now the semigroup ( R, ·) deﬁned by Table 1: Table 1. The Cayley table of the semigroup ( R, ·)
·
0
1
2
3
0 1 2 3
0 0 0 0
0 1 1 1
0 2 2 2
0 3 3 3
Deﬁne on R the hyperoperation +≤ as follows: x +≤ y = {z  x ≤ z ∨ y ≤ z}, so its Cayley table is described in Table 2: Table 2. The Cayley table of the hypergroupoid ( R, +≤ )
+≤
0
1
2
3
0 1 2 3
R R R R
R {1,2,3} {1,2,3} {1,2,3}
R {1,2,3} {2,3} {2,3}
R {1,2,3} {2,3} {3}
Obviously, the relation ≤ is reﬂexive and transitive and, for all x, y, z ∈ R, it holds: x ≤ y ⇒ z · x ≤ z · y. Thus, ( R, +≤ , ·) is an (additive) hypernearring. Let H = R ∪ {4} and deﬁne the hyperoperation +≤ as follows: ( x +≤ y =
x +≤ y, H,
35
if x, y ∈ {0, 1, 2, 3} if x = 4 ∨ y = 4
Symmetry 2019, 11, 964
It follows that ( R, +) is a proper subhypergroup of ( H, +) and for a ﬁxed r ∈ R it holds f r ( x ) = r, for all x ∈ H. This implies that the mapping μ : H → P∗ ( H ), deﬁned by μ(r ) = f r for any r ∈ R, is an inclusive homomorphism. Now we will construct a left general additive hypernearring associated with an arbitrary hypergroup. Theorem 6. Let ( H, +) be a hypergroup and T ( H ) = { f : H → H }. On the set T ( H ) deﬁne the hyperoperation ⊕ T and the operation T as follows: f ⊕ T g = {h ∈ T ( H )  (∀ x ∈ H ) h( x ) ∈ f ( x ) + g( x )},
( f T g)( x ) = g( f ( x )), for all x ∈ H. The obtained structure ( T ( H ), ⊕ T , T ) is a (left) general additive hypernearring. Proof. Let f , g ∈ T ( H ). We prove that there exists h ∈ T ( H ) such that h( x ) ∈ f ( x ) + g( x ) for all x ∈ H. Let x ∈ H. Since f ( x ) + g( x ) = ∅ we can choose h x ∈ f ( x ) + g( x ) and deﬁne h( x ) = h x . Obviously, h ∈ f ⊕ T g. Now we prove that the hyperoperation ⊕ T is associative. Let f , g, h ∈ T ( H ). Set L = ( f ⊕ T g) ⊕ T h = {h  (∀ x ) h ( x ) ∈ h ( x ) + h( x ) ∧ h ( x ) ∈ f ( x ) + g( x )} and D = f ⊕ T ( g ⊕ T h) = { f  (∀ x ) f ( x ) ∈ f ( x ) + f ( x ) ∧ f ( x ) ∈ g( x ) + h( x )}. Thus, if h ∈ L, then h ( x ) ∈ ( f ( x ) + g( x )) + h( x ) = f ( x ) + ( g( x ) + h( x )). Thereby, for any x ∈ H, there exists a x ∈ g( x ) + h( x ) such that h ( x ) ∈ f ( x ) + a x . Deﬁne f ( x ) = a x . Then, f ∈ g ⊕ T h and for all x ∈ H it holds h ( x ) ∈ f ( x ) + f ( x ). Therefore, h ∈ D. So, L ⊆ D. Similarly, we obtain that D ⊆ L. Now, let f , g ∈ T ( H ). We prove that the equation f ∈ g ⊕ T h has a solution h ∈ T ( H ). Since ( H, +) is a hypergroup, it follows that, for any x ∈ H, there exists bx ∈ H such that f ( x ) ∈ g( x ) + bx . Deﬁne h : H → H by h( x ) = bx . Then h ∈ T ( H ) and f ∈ g ⊕ T h. Similarly, we obtain that the equation f ∈ h ⊕ T g has a solution in T ( H ). We may conclude that ( T ( H ), ⊕ T ) is a hypergroup. Obviously, ( T ( H ), T ) is a semigroup, because the composition of functions is associative. Now we prove that the hyperoperation ⊕ T is left inclusively distributive with respect to the operation T . Let f , g, h ∈ T ( H ). Set L = f T ( g ⊕ T h) = { f T k  k ∈ g ⊕ T h} and D = ( f T g) ⊕ T ( f T h) = { h  (∀ x ∈ H ) h ( x ) ∈ g( f ( x )) + h( f ( x ))}. Let k ∈ g ⊕ h. Then, for all x ∈ H, it holds ( f k )( x ) = k( f ( x )) ⊆ g( f ( x )) + h( f ( x )). Thus, f k ∈ D, meaning that L ⊆ D. For an arbitrary group G, Malone and Heatherly [2] denote by T0 ( G ) the subset of T ( G ) consisting of the functions which commute multiplicatively with the zerofunction, i.e., T0 ( G ) = { f : G → G  f (0) = 0}. Obviously, T0 ( G ) is a subnearring of ( T ( G ), +, ·). The next result extends this property to the case of hyperstructures. Theorem 7. Let ( H, +) be a hypergroup with the identity element 0 (i.e., for all x ∈ H, it holds x ∈ x + 0 ∩ 0 + x), such that 0 + 0 = {0}. Let T0 ( H ) = { f : H → H  f (0) = 0}. Then, T0 ( H ) is a subhypernearring of the general additive hypernearring ( T ( H ), ⊕ T , T ). Proof. Let f , g ∈ T0 ( H ). If h ∈ f ⊕ T g, then h(0) ∈ f (0) + g(0) = 0 + 0 = {0}, i.e., h(0) = 0. Thus, h ∈ T0 ( H ). Let f , g ∈ T0 ( H ). We prove now that the equation f ∈ g ⊕ a has a solution a ∈ T0 ( H ). If we set a(0) = 0 and a( x ) = a x , where f ( x ) ∈ g( x ) + a x , for x = 0 and a x ∈ H, then a ∈ T0 ( H ) and f ∈ g + a. Similarly the equation f ∈ a ⊕ g has a solution a ∈ T0 ( H ). Thus, ( T0 ( H ), ⊕ T ) is a subhypergroup of ( T ( H ), ⊕ T ). Obviously, if f , g ∈ T0 ( H ), then it follows that ( f T g)(0) = g( f (0)) = g(0) = 0, i.e., f T g ∈ T0 ( H ). So, ( T0 ( H ), T ) is a subsemihypergroup of ( T ( H ), T ), implying that T0 ( H ) is a subsemihypernearring of T ( H ).
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Symmetry 2019, 11, 964
Theorem 8. Let ( R, +, ·) be an additive hypernearring such that ( R, +) is a proper subhypergroup of the hypergroup ( H, +), having an identity element 0 satisfying the following properties: 1. 2.
0 + 0 = {0} and 0 · r = 0, for all r ∈ R.
Then the hypernearring ( R, +, ·) is W −embeddable in the additive hypernearring T0 ( H ). Proof. For a ﬁxed r ∈ R, deﬁne a map f : H → H as follows (
g · r, if g ∈ R r, if g ∈ H \ R.
f r ( g) =
Obviously, f r (0) = 0 · r = 0. So, f r ∈ T0 ( H ) and, similarly as in the proof of Theorem 5, we obtain that the map ρ : ( R, +, ·) → ( T0 ( H ), ⊕ T , T ) deﬁned by ρ(r ) = f r is an injective inclusion homomorphism. Example 4. On the set H = {0, 1, 2, 3, 4, 5, 6} deﬁne an additive hyperoperation and a multiplicative operation having the Cayley tables described in Tables 3 and 4, respectively: Table 3. The Cayley table of the hypergroupoid ( H, +) +
0
1
2
3
4
5
6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
1 2 3 4 5 {0, 6} 1
2 3 4 5 {0, 6} 1 2
3 4 5 {0, 6} 1 2 3
4 5 {0, 6} 1 2 3 4
5 {0, 6} 1 2 3 4 5
6 1 2 3 4 5 0
Table 4. The Cayley table of the semigroup ( H, ·)
·
0
1
2
3
4
5
6
0 1 2 3 4 5 6
0 0 0 0 0 0 0
0 5 1 0 5 1 0
0 4 2 0 4 2 0
0 3 3 0 3 3 0
0 2 4 0 2 4 0
0 1 5 0 1 5 0
0 0 0 0 0 0 0
The structure ( H, +, ·) is an (additive) hypernearring [16]. Let R = {0, 3, 6}. Then ( R, +, ·) is a hypernearring (in particular it is a subhypernearring of ( H, +, ·)). Obviously, ( R, +) is a proper subhypergroup of the hypergroup ( H, +), which has the identity 0 such that 0 + 0 = {0} and 0 · r = 0, for all r ∈ R. It follows that, for each r ∈ {0, 3, 6}, f r : H → H is a map such that f 0 ( g) = 0, ( for all g ∈ H, ( g · 3, if g ∈ {0, 3, 6} 0, if g ∈ {0, 3, 6} = f 3 ( g) = 3, if g ∈ {1, 2, 4} 3, if g ∈ {1, 2, 4}, while ( ( g · 6, if g ∈ {0, 3, 6} 0, if g ∈ {0, 3, 6} = f 6 ( g) = 6, if g ∈ {1, 2, 4} 6, if g ∈ {1, 2, 4}.
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Symmetry 2019, 11, 964
Clearly, the map ρ : ( R, +, ·) → ( T0 ( H ), ⊕ T , T ), deﬁned by ρ(r ) = f r , is an injective inclusion homomorphism, so the hypernearring R is Wembeddable in T0 ( H ). Remark 3. If ( G, +) is a group, then, for any f , g ∈ T ( G ) = { f : G → G }, it holds f ⊕ T g = f + g and f T g = f · g, meaning that the transformation nearring ( T ( G ), +, ·) of a group G is in fact the structure ( T ( G ), ⊕ T , T ). Furthermore, if ( R, +, ·) is a zerosymmetric nearring, i.e., a nearring in which any element x satisﬁes the relation x · 0 = 0 · x = 0, then the map ρ constructed in the proof of Theorem 8 is the injective homomorphism ρ : R → T0 ( G ). Thus, according with Theorem 8, it follows that the zerosymmetric nearring ( R, +, ·) is Wembeddable in the nearring T0 ( G ), where ( G, +) is any group containing ( R, +) as a proper subgroup. Remark 4. If ( G, +) is a group, then the following inclusions hold: T0 ( G ) ⊆ T ( G ) ⊆ T ∗ ( G ), where both T ( G ) and T0 ( G ) are sub(hyper)nearrings of the hypernearring T ∗ ( G ). Considering now ( H, +) a hypergroup, the same inclusions exist: T0 ( H ) ⊆ T ( H ) ⊆ T ∗ ( H ), but generally T ( H ) and T0 ( H ) are not subhypernearrings of T ∗ ( H ). Proposition 1. Let ( H, +) be a hypergroup with the identity element 0 (i.e., for all x ∈ H it holds x ∈ x + 0 ∩ 0 + x) such that 0 + 0 = {0}. Let T0∗ ( H ) = { f : H → P∗ ( H )  f (0) = 0}. Then, T0∗ ( H ) is a subhypernearring of the general hypernearring ( T ∗ ( H ), ⊕, ). Proof. Let f , g ∈ T0∗ ( H ). If h ∈ f ⊕ g, then it holds h(0) ⊆ f (0) + g(0) = 0 + 0 = {0}. Since h(0) = ∅, it follows that h(0) = {0}. Thus, h ∈ T0∗ ( H ). Let f , g ∈ T0∗ ( H ). We prove that the equation f ∈ g ⊕ a has a solution a ∈ T0∗ ( H ). If we set a(0) = 0 and a( x ) = H, for all x = 0, then a ∈ T0∗ ( H ) and, for all x = 0, it holds g( x ) + a( x ) = H ⊇ f ( x ) and g(0) + a(0) = {0} = f (0), meaning that f ∈ g ⊕ a. Similarly, the equation f ∈ a ⊕ g has a solution in T0∗ ( H ). So, ( T0∗ ( H )) is a subhypergroup of ( T ∗ ( H ), ⊕). Obviously, if h ∈ f g, then h(0) ⊆ g( f (0)) = {0}. So, h ∈ T0∗ ( H ). Thus T0∗ ( H ) is a subsemihypergroup of ( T ∗ ( H ), ). Therefore, T0∗ ( H ) is a subhypernearring of ( T ∗ ( H ), ⊕, ). 4. Conclusions Distributivity property plays a fundamental role in the ringlike structures, i.e., algebraic structures endowed with two operations, usually denoted by addition and multiplication, where the multiplication distributes over the addition. If this happens only from onehand side, then we talk about nearrings. Similarly, in the framework of algebraic hypercompositional structures, a general hypernearring has the additive part an arbitrary hypergroup, the multiplicative part is a semihypergroup, and the multiplication hyperoperation inclusively distributes over the hyperaddition from the left or righthand side, i.e., for three arbitrary elements x, y, z, there is x · (y + z) ⊆ x · y + x · z for the lefthand side, and respectively, (y + z) · x ⊆ y · x + z · x for the righthand side. If the inclusion is substituted by equality, then the general hypernearring is called strongly distributive. We also recall here that there exist also hyperrings having the additive part a group, while the multiplicative one is a semihypergroup, being called multiplicative hyperrings [20]. The set of all transformations of a group G, i.e., T ( G ) = { g : G → G }, can be endowed with a nearring structure, while similarly, on the set of all multitransformations of a hypergroup H, i.e., T ∗ ( H ) = { h : H → P ∗ ( H )}, can be deﬁned a general hypernearring structure, called the multitransformations general hypernearring associated with the hypergroup H. We have shown that for every general hypernearring R there exists a hypergroup H such that R is weakly embeddable in the associated multitransformations general hypernearring T ∗ ( H ) (see Theorem 5). Moreover, considering the set T ( H ) = { f : H → H } of all transformations of a hypergroup H, we have deﬁned on it a hyperaddition and a multiplication such that T ( H ) becomes a general additive hypernearring. We have determined conditions under which the set T0 ( H ), formed with the transformations of H that multiplicatively commute with the zero function on H, is a subhypernearring of T ( H ). Besides,
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an additive hypernearring satisfying certain conditions can be weakly embedded in the additive hypernearring T0 ( H ) (see Theorem 8). In our future work, we intend to introduce and study properties of Δ−endomorphisms and Δ−multiendomorphisms of hypernearrings as generalizations of similar notions on nearrings. Author Contributions: The authors contributed equally to this paper. Funding: The third author acknowledges the ﬁnancial support from the Slovenian Research Agency (research core funding No. P10285). Conﬂicts of Interest: The authors declare no conﬂict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Berman, G.; Silverman, R.J. Nearrings. Am. Math. Mon. 1959, 66, 23–34. [CrossRef] Malone, J.J.; Heatherly, H.E., Jr. Some NearRing Embeddings. Quart. J. Math. Oxf. Ser. 1969, 20, 81–85. [CrossRef] Daši´c, V. Hypernearrings. In Algebraic Hyperstructures and Applications (Xanthi, 1990); World Scientiﬁc Publishing: Teaneck, NJ, USA, 1991; pp. 75–85. Bonansinga, P. Sugli ipergruppi quasicanonici. Atti Soc. Peloritana Sci. Fis. Mat. Natur 1981, 27, 9–17. Massouros, C.G. Quasicanonical hypergroups. In Algebraic Hyperstructures and Applications (Xanthi, 1990); World Scientiﬁc Publishing: Teaneck, NJ, USA, 1991; pp. 129–136. Comer, S.D. Polygroups derived from cogroups. J. Algebra 1984, 89, 387–405. [CrossRef] Davvaz, B. Polygroup Theory and Related Systems; World Scientiﬁc Publishing, Co. Pte. Ltd.; Hackensack, NJ, USA, 2013. Janˇci´cRašovi´c, S.; Cristea, I. A note on nearrings and hypernearrings with a defect of distributivity. AIP Conf. Proc. 1978, 1978, 34007. Ameri, R.; AmiriBideshki, M.; HoskovaMayerova, S.; Saeid, A.B. Distributive and Dual Distributive Elements in Hyperlattices. Ann. Univ. Ovidius Constanta Ser. Mat. 2017, 25, 25–36. [CrossRef] Janˇci´cRašovi´c, S.; Cristea, I. Division hypernearrings. Ann. Univ. Ovidius Constanta Ser. Mat. 2018, 26, 109–126. [CrossRef] Gontineac, M. On Hypernearring and Hhypergroups. In Algebraic Hyperstructures and Applications (Lasi, 1993); Hadronic Press: Palm Harbor, FL, USA, 1994; pp. 171–179. Daši´c, V. A defect of distributivity of the nearrings. Math. Balk. 1978, 8, 63–75. Clay, J. Nearrings: Geneses and Application; Oxford University Press: Oxford, UK, 1992. Meldrum, J. NearRings and Their Links with Groups; Pitman: London, UK, 1985. Pilz, G. NearRings: The theory and Its Applications; NorthHolland Publication Co.: New York, NY, USA, 1983. Janˇci´cRašovi´c, S.; Cristea, I. Hypernearrings with a defect of distributivity. Filomat 2018, 32, 1133–1149. [CrossRef] Vougiouklis, T. Hyperstructures and Their Representations; Hadronic Press: Palm Harbor, FL, USA, 1994. Janˇci´cRašovi´c, S. On a class of P1 − P2 hyperrings and hypernearrings. SetVal. Math. Appl. 2008, 1, 25–37. Janˇci´cRašovi´c, S.; Dasic, V. Some new classes of (m, n)hyperrings. Filomat 2012, 26, 585–596. [CrossRef] Ameri, R.; Kordi, A.; HoškovaMayerova, S. Multiplicative hyperring of fractions and coprime hyperideals. Ann. Univ. Ovidius Constanta Ser. Mat. 2017, 25, 5–23. [CrossRef] c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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SS symmetry Article
Edge Even Graceful Labeling of Cylinder Grid Graph Ahmed A. Elsonbaty 1,2 and Salama Nagy Daoud 1,3, * 1 2 3
*
Department of Mathematics, Faculty of Science, Taibah University, AlMadinah 41411, Saudi Arabia; [email protected] Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt Department of Mathematics and Compuer Science, Faculty of Science, Menouﬁa University, Shebin El Kom 32511, Egypt Correspondence: [email protected]
Received: 26 March 2019; Accepted: 14 April 2019; Published: 22 April 2019
Abstract: Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An graph G = (V (G), E(G)) of e.e.g., l. of simple ﬁnite undirected order P = (V (G) and size q = E(G) is a bijection f : E(G) → 2, 4, . . . , 2q , such that when each vertex v ∈ V (G) is assigned the modular sum of the labels (images of f ) of the edges incident to v, the resulting vertex labels are distinct mod2r, where r = max(p, q). In this work, the family of cylinder grid graphs are studied. Explicit formulas of e.e.g., l. for all of the cases of each member of this family have been proven. Keywords: graceful labeling; edge even graceful labeling; cylinder grid graph
1. Introduction The ﬁeld of graph theory plays an important role in various areas of pure and applied sciences. One of the important areas in graph theory is graph labeling of a graph G which is an assignment of integers either to the vertices or edges or both subject to certain conditions. Graph labeling began nearly 50 years ago. Over these decades, more than 200 methods of labeling techniques were invented, and more than 2500 papers were published. In spite of this huge literature, just few general results were discovered. Nowadays, graph labeling has much attention from diﬀerent brilliant researchers in graph theory, which has rigorous applications in many disciplines, e.g., communication networks, coding theory, Xray crystallography, radar, astronomy, circuit design, communication network addressing, database management, and graph decomposition problems. More interesting applications of graph labeling can be found in References [1–11]. A function f is called a graceful labeling of a graph G if f : V (G) → 0, 1, 2, . . . , q is injective and the induced function f ∗ : E(G) → 1, 2, . . . , q , deﬁned as ∗ f (e = uv) = f (u) − f (v) , is bijective. This type of graph labeling was ﬁrst introduced by Rosa in 1967 [12] as a β− valuation, and later, Solomon W. Golomb [13] termed it as graceful labeling. A function f is called an odd graceful labeling of a graph G if f : V (G) → 0, 1, 2, . . . , 2q − 1 is injective and the ∗ ∗ induced function f : E(G) → 1, 3, . . . , 2q − 1 , deﬁned as f (e = uv) = f (u) − f (v), is bijective. This type of graph labeling ﬁrst introduced by Gnanajothi in 1991 [14]. For more results on this type of labeling, see References [15,16]. A function f is called an edge graceful labeling of a graph G if f : E(G) → 1, 2, . . . , q is bijective and the induced function f ∗ : V (G) → 0, 1, 2, . . . , p − 1 , deﬁned ∗ as f (u) = f (e)(modp), is bijective. This type of graph labeling was ﬁrst introduced by e=uv∈E(G)
Lo in 1985 [17]. For more results on this labeling see [18,19]. A function f is called an edge odd graceful labeling of a graph G if f : E(G) → 1, 3, . . . , 2q − 1 is bijective and the induced function f ∗ : V (G) → 0, 1, 2, . . . , 2q − 1 deﬁned as f ∗ (u) = f (e)(mod2q) is injective. This type of e=uv∈E(G)
graph labeling was ﬁrst introduced by Solairaju and Chithra in 2009 [20]. For more results on this
Symmetry 2019, 11, 584; doi:10.3390/sym11040584
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Symmetry 2019, 11, 584
labeling, see References [21–23]. A function f is called an edge even graceful labeling of a graph G if f : E(G) → 2, 4, . . . , 2q − 2 is bijective and the induced function f ∗ : V (G) → 0, 2, 4, . . . , 2q − 2 , ∗ deﬁned as f (u) = f (e)(mod2r) where r = max p, q , is injective. This type of graph labeling e=uv∈E(G)
was ﬁrst introduced by Elsonbaty and Daoud in 2017 [24,25]. For a summary of the results on these ﬁve types of graceful labels as well as all known labeling techniques, see Reference [26]. 2. Cylinder Grid Graph The Cartesian product G1 × G2 of two graphs G1 and G2 , is the graph with vertex set V (G1 ) × V (G2 ), and any two vertices (u1 , v1 ) and (u2 , v2 ) are adjacent in G1 × G2 whenever u1 = u2 and v1 v2 ∈ E(G2 ) or v1 = v2 and u1 u2 ∈ E(G1 ). The cylinder grid graph Cm,n is the graph formed from the Cartesian product Pm × Cn of the path graph Pm and the cycle graph Cn . That is, the cylinder grid graph consists of m copies of Cn represented by circles, and will be numbered from the innermost circle to the outer (1)
(2)
(3)
(m−1)
circle as Cn , Cn , Cn , . . . , Cn
(m)
, Cn
and we call them simply circles; n copies of Pm represented by (1)
(2)
(3)
(n−1)
paths transverse the m circles and will be numbered clockwise as Pm , Pm , Pm , . . . , Pm call them paths (see Figure 1).
(n)
, Pm and we
Figure 1. Cylinder grid graph Cm,n .
Theorem 1. If m is an even positive integer greater than or equal 2 and n ≥ 2, then the cylinder grid graph Cm,n , is an edge even graceful graph.
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Symmetry 2019, 11, 584
Proof. Using standard notation p = V(Cm,n ) = mn, q = E(Cm,n ) = 2mn − n and r = max(p, q) = 2mn − n and f : E(Cm,n ) → {2, 4, 6, . . . , 4mn − 2n − 2}. Let the cylinder grid graph Cm,n be as in Figure 2.
Figure 2. The cylinder grid graph Cm,n , m is even and n ≥ 2. (k )
(1) Pm
First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path
as follows: Move anticlockwise to label the edges v1 vn+1 , vn v2n , vn−1 v2n−1 , . . . , v3 vn+3 , v2 vn+2 by 2, 4, 6, . . . , 2n − 2, 2n, then move clockwise to label the edges vn+1 v2n+1 , vn+2 v2n+2 , vn+3 v2n+3 , . . ., v2n−1 v3n−1 , v2n v3n by 2n + 2, 2n + 4, 2n + 6, . . . , 4n − 2, 4n, then move anticlockwise to label the edges v2n+1 v3n+1 , v3n v4n , v3n−1 v4n−1 , . . . , v2n+3 v3n+3 , v2n+2 v3n+2 by 4n + 2, 4n + 4, 4n + 6, . . . , 6n − 2, 6n and so on. Finally, move anticlockwise to label the edges v(m−2)n+1 v(m−1)n+1 , v(m−1)n vmn , v(m−1)n−1 vmn−1 , . . ., v(m−2)n+3 v(n−1)m+3 , v(m−2)n+2 v(m−1)n+2 by 2n(m − 1) + 2, 2n(m − 2) + 4, 2n(m − 2) + 6, 2n(m − 2) + 8, . . . , 2n(m − 1) − 2, 2n(m − 1). (k )
Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost (1) Cn
(m) (m−2) (m−4) (2) circle then the edges of outer circle Cn , then the edges of the circles Cn , Cn , . . . , Cn . (m−1) (m−3) (3) Finally, we label the edges of the circles Cn , Cn , . . . , Cn as follows: f (vi vi+1 ) = 2n(m − 1) +
2i, 1 ≤ i ≤ n − 1, f (vn v1 ) = 2mn; f (v(m−1)n+i v(m−1)n+i+1 ) = 2mn + 2i, 1 ≤ i ≤ n − 1, f (vmn v(m−1)n+1 ) = 2n(m + 1); f (v(k−1)n+i v(k−1)n+i+1 ) = n(3m − k) + 2i, 1 ≤ i ≤ n − 1, f (vkn v(k−1)n+1 ) = n(3m − k + 2), 2 ≤ k ≤ m − 2; f (v(k−1)n+i v(k−1)n+i+1 ) = n(4m − k − 1) + 2i, 1 ≤ i ≤ n − 1, f (vkn v(k−1)n+1 ) = n(4m − k + 1), 3 ≤ k ≤ m − 1, k is odd. 42
Symmetry 2019, 11, 584
Thus, the labels of corresponding vertices mod(4mn − 2n) will be: f ∗ (vi ) ≡ 2i + 2; f ∗ (vn+i ) ≡ 2mn + 2n + 4i + 2; f ∗ (v2n+i ) ≡ 4n + 4i + 2; f ∗ (v3n+i ) ≡ 2mn + 6n + 4i + 2; . . . ; f ∗ (v(m−3)n+i ) ≡ 4mn − 6n + 4i + 2; f ∗ (v(m−2)n+i ) ≡ 2mn − 4n + 4i + 2; f ∗ (v(m−1)n+i ) ≡ 2mn + 2i + 2, 1 ≤ i ≤ n. Illustration: An e.e.g., l, of the cylinder grid graphs C8,11 and C8,12 are shown in Figure 3. Theorem 2. If m = 3 and n is an odd positive integer greater than 3, then the cylinder grid graph C3,n , is an edge even graceful graph. Proof. Using standard notation p = V (C3,n ) = 3n, q = E(C3,n ) = 5n, r = max(p, q) = 5n, and f : E(C3,n ) → {2, 4, 6, . . . , 10n − 2}. There are three cases: Case (1): If n ≡ 1mod6, let the cylinder grid graph C3,n be as in Figure 4.
(a) & Figure 3. Cont.
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(b) & Figure 3. An edge even graceful labeling (e.e.g., l.) of the cylinder grid graphs C8,11 and C8,12 . (k )
(1)
First, we label the edges of the paths P3 , 1 ≤ k ≤ n beginning with the edges of the path
P3 as follows: Move clockwise to label the edges f (v1 vn+1 ) = 2, f (v2 vn+2 ) = 6, f (vi vn+i ) = 2i + 2, 3 ≤ i ≤ n. Then, move anticlockwise to label the edges f (vn+1 v2n+1 ) = 2n + 4, f (v2n v3n ) = 2n + 6, f (v2n−1 v3n−1 ) = 2n + 8, f (v2n−2 v3n−2 ) = 2n + 10, . . . , f (vn+3 v2n+3 ) = 4n, f (vn+2 v2n+2 ) = 4n + 2. (k )
Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ 3 beginning with the edges of the innermost
(2) then the edges of outer circle and then the edges of the circle Cn . Label the edges (1) the circle Cn as follows: f (v1 v2 ) = 4n + 4, f (v2 v3 ) = 4n + 6, . . . , f (v n−1 v n+2 ) = 14n3+10 , f (v n+2 v n+5 ) 3 3 3 3 14n+4 14n+16 , f (v n+8 v n+11 ) = 14n3+22 , f (v n+11 v n+14 ) = 14n3+34 , f (v n+14 v n+17 ) 5 v n+8 ) = 3 , f ( v n+ 3 3 3 3 3 3 3 3 3 14n+28 , f (v n+17 v n+20 ) = 14n3+40 , f (v n+20 v n+23 ) = 14n3+46 , f (v n+23 v n+26 ) = 14n3+58 , f (v n+26 v n+29 ) 3 3 3 3 3 3 3 3 3 14n+52 , f (v n+29 v n+32 ) = 14n3+64 , f (v n+32 v n+35 ) = 14n3+70 , f (v n+35 v n+38 ) = 14n3+82 , f (v n+38 v n+41 ) 3 3 3 3 3 3 3 3 3 14n+76 , f (v n+41 v n+44 ) = 14n3+88 , f (v n+44 v n+47 ) = 14n3+94 , . . . , f (vn−13 vn−12 ) = 6n − 22, f (vn−12 vn−11 ) 3 3 3 3 3
circle
(1) Cn ,
(3) Cn ,
of
= = = =
= 6n − 24, f (vn−11 vn−10 ) = 6n − 20, f (vn−10 vn−9 ) = 6n − 18, f (vn−9 vn−8 ) = 6n − 14, f (vn−8 vn−7 ) = 6n − 16, f (vn−7 vn−6 ) = 6n − 12, f (vn−6 vn−5 ) = 6n − 10, f (vn−5 vn−4 ) = 6n − 6, f (vn−4 vn−3 ) = 6n − 8, f (vn−3 vn−2 ) = 6n − 4, f (vn−2 vn−1 ) = 6n − 2, f (vn−1 vn ) = 6n + 2, f (vn v1 ) = 6n.
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Symmetry 2019, 11, 584
Figure 4. The cylinder grid graph C3,n , n ≡ 1mod6. (2)
Label the edges of the circle Cn as follows: f (vn+i vn+i+1 ) = 8n + 2i + 2, 1 ≤ i ≤ n − 1, f (v2n vn+1 ) = 4. (3)
Label the edges of the circle Cn as follows: f (v2n+i v2n+i+1 ) = 6n + 2i + 2, 1 ≤ i ≤ n. The labels of corresponding of vertices mod10n are as follows: (1)
The labels of vertices of the circle Cn are as follows: f ∗ (v1 ) ≡ 6, f ∗ (v2 ) ≡ 8n + 16, f ∗ (v3 ) ≡ 8n + 22, . . . , f ∗ (v n−1 ) ≡ 4, f ∗ (v n+2 ) ≡ 8, f ∗ (v n+5 ) ≡ 12, f ∗ (v n+8 ) ≡ 20, f ∗ (v n+11 ) ≡ 28, f ∗ (v n+14 ) ≡ 3
3
3
3
3
3
32, f ∗ (v n+17 ) ≡ 36, f ∗ (v n+20 ) ≡ 44, f ∗ (v n+23 ) ≡ 52, f ∗ (v n+26 ) ≡ 56, f ∗ (v n+29 ) ≡ 60, f ∗ (v n+32 ) ≡ 68, 3
3
3
3
3
3
f ∗ (v n+35 ) ≡ 76, f ∗ (v n+38 ) ≡ 80, f ∗ (v n+41 ) ≡ 84, f ∗ (v n+44 ) ≡ 92, f ∗ (v n+47 ) ≡ 100, . . . , f ∗ (vn−12 ) ≡ 4n − 3
3
3
3
3
68, f ∗ (vn−11 ) ≡ 4n − 64, f ∗ (vn−10 ) ≡ 4n − 56, f ∗ (vn−9 ) ≡ 4n − 48, f ∗ (vn−8 ) ≡ 4n − 44, f ∗ (vn−7 ) ≡ 4n − 40, f ∗ (vn−6 ) ≡ 4n − 32, f ∗ (vn−5 ) ≡ 4n − 24, f ∗ (vn−4 ) ≡ 4n − 20, f ∗ (vn−3 ) ≡ 4n − 16, f ∗ (vn−2 ) ≡ 4n − 8, f ∗ (vn−1 ) ≡ 4n, f ∗ (vn ) ≡ 4n + 4. (2)
The labels of vertices of the circle Cn are f ∗ (vi+1 ) = 4i + 10, 1 ≤ i ≤ n − 1, f ∗ (v2n ) = 4n + 12. (3)
The labels of vertices of the circle Cn are f ∗ (v2i+1 ) = 6n + 2i + 8, 1 ≤ i ≤ n. Case (2): If n ≡ 3mod6, let the cylinder grid graph C3,n be as in Figure 5. (k )
(1)
First, we label the edges of the paths P3 , 1 ≤ k ≤ n beginning with the edges of the path P3 as the same in case (1). (k ) Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ 3 beginning with the edges of the innermost (1)
(3)
(2)
circle Cn , then the edges of outer circle Cn , and then the edges of the circle Cn . 45
Symmetry 2019, 11, 584
Figure 5. The cylinder grid graph C3,n , n ≡ 3mod6. (1)
Label the edges of the circle Cn as follows: f (v1 v2 ) = 4n + 4, f (v2 v3 ) = 4n + 6, . . . , f (v n+3 v n+6 ) = 14n+18 , 3 14n+42 , 3 14n+66 , 3 14n+90 , 3
f ( v n+6 v n+9 ) = 3
3
f (v n+18 v n+21 ) = 3
3
3
3
f (v n+30 v n+33 ) = f (v n+42 v n+45 ) =
3
3
14n+12 , f (v n+9 v n+12 ) = 14n3+24 , f (v n+12 v n+15 ) = 14n3+30 , f (v n+15 v n+18 ) 3 3 3 3 3 3 3 14n+36 , f (v n+21 v n+24 ) = 14n3+48 , f (v n+24 v n+27 ) = 14n3+54 , f (v n+27 v n+30 ) 3 3 3 3 3 3 3 14n+60 , f (v n+33 v n+36 ) = 14n3+72 , f (v n+36 v n+39 ) = 14n3+78 , f (v n+39 v n+42 ) 3 3 3 3 3 3 3 14n+84 102 , f (v n+45 v n+48 ) = 14n3+96 , f (v n+48 v n+51 ) = 14n+ , . . ., f (vn−13 vn−12 ) 3 3 3 3 3 3
= = =
= 3 3 6n − 22, f (vn−12 vn−11 ) = 6n − 24, f (vn−11 vn−10 ) f (vn−9 vn−8 ) = 6n − 14, f (vn−8 vn−7 ) = 6n − 16, f (vn−7 vn−6 ) = 6n − 12, f (vn−6 vn−5 ) = 6n − 10, f (vn−5 vn−4 ) = 6n − 6, f (vn−4 vn−3 ) = 6n − 8, f (vn−3 vn−2 ) = 6n − 4, f (vn−2 vn−1 ) = 6n − 2, f (vn−1 vn ) = 6n + 2, f (vn v1 ) = 6n. (1) The labels of corresponding vertices mod10n are as follows: The label of vertices of the circle Cn are f ∗ (v1 ) ≡ 6, f ∗ (v2 ) ≡ 8n + 16, f ∗ (v3 ) ≡ 8n + 22, . . . , f ∗ (v n −1 ) ≡ 10n − 2, f ∗ (v n ) ≡ 4, f ∗ (v n +1 ) ≡ 3 3 3 12, f ∗ (v n +2 ) ≡ 16, f ∗ (v n +3 ) ≡ 20, f ∗ (v n +4 ) ≡ 28, f ∗ (v n +5 ) ≡ 36, f ∗ (v n +6 ) ≡ 40, f ∗ (v n +7 ) ≡ 3 3 3 3 3 3 44, f ∗ (v n +8 ) ≡ 52, f ∗ (v n +9 ) ≡ 60, f ∗ (v n +10 ) ≡ 64, f ∗ (v n +11 ) ≡ 68, f ∗ (v n +12 ) ≡ 76, f ∗ (v n +13 ) ≡ 3 3 3 3 3 3 84, f ∗ (v n +14 ) ≡ 88, f ∗ (v n +15 ) ≡ 92, f ∗ (v n +16 ) ≡ 100, . . . , f ∗ (vn−12 ) ≡ 4n − 68, f ∗ (vn−11 ) ≡ 4n − 64, 3 3 3 f ∗ (vn−10 ) ≡ 4n − 56, f ∗ (vn−9 ) ≡ 4n − 48, f ∗ (vn−8 ) ≡ 4n − 44, f ∗ (vn−7 ) ≡ 4n − 40, f ∗ (vn−6 ) ≡ 4n − 32, f ∗ (vn−5 ) ≡ 4n − 24, f ∗ (vn−4 ) ≡ 4n − 20, f ∗ (vn−3 ) ≡ 4n − 16, f ∗ (vn−2 ) ≡ 4n − 8, f ∗ (vn−1 ) ≡ 4n, f ∗ (vn ) ≡ 4n + 4. (2) (3) The labels of vertices of the circles Cn and Cn are the same as in case (1). Case (3): If n ≡ 5mod6, let the cylinder grid graph C3,n be as in Figure 6.
46
Symmetry 2019, 11, 584
Figure 6. The cylinder grid graph C3,n , n ≡ 5mod6. (k )
(1)
First, we label the edges of the paths P3 , 1 ≤ k ≤ 2 beginning with the edges of the path P3 as
(k ) the same in case (1). Second, we label the edges of the circles Cn , 1 ≤ k ≤ 3 beginning with the edges (1) (3) (2) of the innermost circle Cn , then the edges of outer circle Cn , and then the edges of the circle Cn . (1) Label the edges of the circle Cn as follows: f (v1 v2 ) = 4n + 4, f (v2 v3 ) = 4n + 6, . . . , f (v n−5 v n−2 ) = 3 3 14n+2 v n+1 ) = 14n−4 , f (v n+1 v n+4 ) = 14n3+8 , f (v n+4 v n+7 ) = 14n3+14 , f (v n+7 v n+10 ) = 3 , f (v n−2 3 3 3 3 3 3 3 3 3 14n+26 , f (v n+10 v n+13 ) = 14n3+20 , f (v n+13 v n+16 ) = 14n3+32 , f (v n+16 v n+19 ) = 14n3+38 , f (v n+19 v n+22 ) = 3 3 3 3 3 3 3 3 3 14n+50 , f (v n+22 v n+25 ) = 14n3+44 , f (v n+25 v n+28 ) = 14n3+56 , f (v n+28 v n+31 ) = 14n3+62 , f (v n+31 v n+34 ) = 3 3 3 3 3 3 3 3 3 14n+74 , f (v n+34 v n+37 ) = 14n3+68 , f (v n+37 v n+40 ) = 14n3+80 , f (v n+40 v n+43 ) = 14n3+86 , . . . , f (vn−13 vn−12 ) = 3 3
3
3
3
3
3
6n − 22, f (vn−12 vn−11 ) = 6n − 24, f (vn−11 vn−10 ) = 6n − 20, f (vn−10 vn−9 ) = 6n − 18, f (vn−9 vn−8 ) = 6n − 14, f (vn−8 vn−7 ) = 6n − 16, f (vn−7 vn−6 ) = 6n − 12, f (vn−6 vn−5 ) = 6n − 10, f (vn−5 vn−4 ) = 6n − 6, f (vn−4 vn−3 ) = 6n − 8, f (vn−3 vn−2 ) = 6n − 4, f (vn−2 vn−1 ) = 6n − 2, f (vn−1 vn ) = 6n + 2, f (vn v1 ) = 6n. The labels of corresponding vertices mod10n are as follows: The labels of vertices of the (1)
circle Cn :
f ∗ (v1 ) ≡ 6, f ∗ (v2 ) ≡ 8n + 16, f ∗ (v3 ) ≡ 8n + 22, . . . , f ∗ (v n −5 ) ≡ 10n − 4, f ∗ (v n−2 ) ≡ 3
3
) ≡ 4, f ∗ (v n+4 ) ≡ 12, f ∗ (v n+7 ) ≡ 20, f ∗ (v n+10 ) ≡ 24, f ∗ (v n+13 ) ≡ 28, f ∗ (v n+16 ) ≡ 36, f ∗ (v n3 +7 ) ≡ 3 3 3 3 3 ∗ 44, f (v n+19 ) ≡ 44, f ∗ (v n+22 ) ≡ 48, f ∗ (v n+25 ) ≡ 52, f ∗ (v n+28 ) ≡ 60, f ∗ (v n+31 ) ≡ 68, f ∗ (v n+34 ) ≡ 3 3 3 3 3 3 72, f ∗ (v n+37 ) ≡ 76, f ∗ (v n+40 ) ≡ 84, f ∗ (v n+43 ) ≡ 92, f ∗ (v n+46 ) ≡ 96, f ∗ (v n+49 ) ≡ 100, . . . , f ∗ (vn−12 ) ≡ 3 3 3 3 3 4n − 68, f ∗ (vn−11 ) ≡ 4n − 64, f ∗ (vn−10 ) ≡ 4n − 56, f ∗ (vn−9 ) ≡ 4n − 48, f ∗ (vn−8 ) ≡ 4n − 44, f ∗ (vn−7 ) ≡ 4n − 40, f ∗ (vn−6 ) ≡ 4n − 32, f ∗ (vn−5 ) ≡ 4n − 24, f ∗ (vn−4 ) ≡ 4n − 20, f ∗ (vn−3 ) ≡ 4n − 16, f ∗ (vn−2 ) ≡ 4n − 8, f ∗ (vn−1 ) ≡ 4n, f ∗ (vn ) ≡ 4n + 4. 0,
f ∗ (v
n+1 3
47
Symmetry 2019, 11, 584
(2)
(3)
The labels of vertices of the circles Cn and Cn are the same as in case (1). Illustration: An e.e.g., l. of the cylinder grid graphs C3,25 , C3,27 and C3,29 are shown in Figure 7.
(a) &
(b) &
Figure 7. Cont.
48
Symmetry 2019, 11, 584
(c) &
Figure 7. An e.e.g., l. of the cylinder grid graphs C3,25 , C3,27 and C3,29 .
Remark 1. Note that C3,5 is an edge even graceful graph but it does not follow the pervious rule (see Figure 8).
Figure 8. An e.e.g., l. of the cylinder grid graph C3,5 .
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Symmetry 2019, 11, 584
Theorem 3. If m is an odd positive integer greater than 3 and n is an even positive integer, n ≥ 2, then the cylinder grid graph Cm,n , is an edge even graceful graph. Proof. Using standard notation p = V (Cm,n ) = mn, q = E(Cm,n ) = 2mn − n and r = max(p, q) = 2mn − n and f : E(Cm,n ) → {2, 4, 6, . . . , 4mn − 2n − 2}. Let the cylinder grid graph Cm,n be as in Figure 9. There are six cases:
Figure 9. The cylinder grid graph Cm,n ,m is odd greater than 3 and n ≥ 2.
Case (1):
n ≡ 0mod12.
(k )
First, we label the edges of the paths Pm , 1 ≤ k ≤ n (1)
beginning with the edges of the path Pm as follows: Move clockwise to label the edges v1 vn+1 , v2 vn+2 , v3 vn+3 , . . . , vn−1 v2n−1 , vn v2n by 2, 4, 6, . . . , 2n − 2, 2n, then move anticlockwise to label the edges vn+1 vn+2 , v2n v3n , v2n−1 v3n−1 , . . . , vn+3 v2n+3 , vn+2 v2n+2 by 2n + 2, 2n + 4, 2n + 6, . . . , 4n − 2, 4n, then move clockwise to label the edges v2n+1 v3n+1 , v2n+2 v3n+2 , v2n+3 v3n+3 , . . . , v3n−1 v4n−1 , v3n v4n by 4n + 2, 4n + 4, 4n + 6, . . . , 6n − 2, 6n and so on. Finally, move anticlockwise to label the edges v(m−2)n+1 v(m−1)n+1 , v(m−1)n vmn , v(m−1)n−1 vmn−1 , . . . , v(m−2)n+3 v(m−1)n+3 , v(m−2)n+2 vm(n−1)+2 by 2n(m − 2) + 2, 2n(m − 2) + 4, 2n(m − 2) + 6, . . . , 2n(m − 1) − 2, 2n(m − 1).
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Symmetry 2019, 11, 584
(k )
Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the inner (1) Cn ,
(m) most circle then the edges of outer circle Cn , then the edges of the circles (m−1) (m−3) (2) Finally, we label the edges of the circles Cm , Cm , . . . , Cm . (1) Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2,
(m−1)
Cm
(m−3)
, Cm
(2)
, . . . , Cm .
f (v2 v3 ) = 2n(m − 1) + 6, f (v3 v4 ) = 2n(m − 1) + 4, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 14, f (v7 v8 ) = 2n(m − 1) + 12, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 22, f (v11 v12 ) = 2n(m − 1) + 20, f (v12 v13 ) = 2n(m − 1) + 24, . . . , f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 2, f (vn−2 vn−1 ) = 2mn − 6, f (vn−1 vn ) = 2mn, f (vn v1 ) = 2mn − 4. (m)
Label the edges of the circle Cn 2i, f (vmn v(m−1)n+1 ) = 2n(m + 1), 1 ≤ i ≤ n − 1.
as follows:
(m−2)
Label the edges of the circle Cn as follows: 2i, f (v(m−2)n v(m−3)n+1 ) = 2n(m + 2), 1 ≤ i ≤ n − 1.
f (v(m−1)n+i v(m−1)n+i+1 )
=
2mn +
f (v(m−3)n+i v(m−3)n+i+1 ) = 2n(m + 1) +
(m−4)
Label the edges of the circle Cn as follows: f (v(m−5)n+i v(m−5)n+i+1 ) = 2n(m + 2) + 2i, f (v(m−4)n v(m−5)n+1 ) = 2n(m + 3), 1 ≤ i ≤ n − 1, and so on. (3)
Label the edges of the circle Cn as follows: f (v2n+i v2n+i+1 ) = 3n(m − 1) + 2i, f (v3n v2n+1 ) = n(3m − 1), 1 ≤ i ≤ n − 1, (m−1)
Label the edges of the circle Cn as follows: 2i, f (v(m−1)n v(m−1)n+1 ) = n(3m + 1) − 1, 1 ≤ i ≤ n − 1,
f (v(m−2)n+i v(m−2)n+i+1 ) = n(3m − 1) +
(m−3)
Label the edges of the circle Cn as follows: f (v(m−4)n+i v(m−4)n+i+1 ) = n(3m + 2) + 2i, f (v(m−3)n v(m−4)n+1 ) = 3n(m + 1), 1 ≤ i ≤ n − 1, . . . , and so on. (4)
Label the edges of the circle Cn as follows: f (v3n+i v3n+i+1 ) = 2n(2m − 3) + 2i, f (v4n v3n+1 ) = 4n(m − 1), 1 ≤ i ≤ n − 1, (2)
Label the edges of Cn as follows: f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, f (v2n v2n+1 ) = 2n(m − 1), 1 ≤ i ≤ n − 1, Thus, the labels of corresponding vertices mod(4mn − 2n) will be: (1)
The label the vertices of Cn are: f ∗ (v1 ) ≡ 0; f ∗ (v2 ) ≡ 4mn − 4n + 12; f ∗ (v3 ) ≡ 4mn − 4n + 16; f ∗ (v4 ) ≡ 4mn − 4n + 20; f ∗ (v5 ) ≡ 4mn − 4n + 28; f ∗ (v6 ) ≡ 4mn − 4n + 36; f ∗ (v7 ) ≡ 4mn − 4n + 40; f ∗ (v8 ) ≡ 4mn − 4n + 44; f ∗ (v9 ) ≡ 4mn − 4n + 52; f ∗ (v10 ) ≡ 4mn − 4n + 60; f ∗ (v11 ) ≡ 4mn − 4n + 64; f ∗ (v12 ) ≡ 4mn − 4n + 68; . . . ; f ∗ (vn−6 ) ≡ 4n − 36; f ∗ (vn−5 ) ≡ 4n − 32; f ∗ (vn−4 ) ≡ 4n − 28; f ∗ (vn−3 ) ≡ 4n − 16; f ∗ (vn−2 ) ≡ 4n − 12; f ∗ (vn−1 ) ≡ 4n − 8; f ∗ (vn ) ≡ 4n − 4. (2)
(3)
(4)
(m−2)
(m−1)
(m)
The label the vertices of Cn , Cn , Cn , . . . , Cn , Cn , Cn respectively are: f ∗ (vn+i ) ≡ 4i + 2; f ∗ (v2n+i ) ≡ 2mn + 4n + 4i + 2; f ∗ (v3n+i ) ≡ 4n + 4i + 2; . . . ; f ∗ (v(m−3)n+i ) ≡ 4mn − 6n + 4i + 2; f ∗ (v(m−2)n+i ) ≡ 2mn − 6n + 4i + 2; f ∗ (v(m−1)n+i ) ≡ 2mn + 2i + 2, 1 ≤ i ≤ n. Case (2): n ≡ 2mod12, n 2. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n begin with the edges of the path Pm as the same in case (1). (k ) Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ m begin with the edges of the inner most (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 6, f (v3 v4 ) = 2n(m − 1) + 4, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 14, f (v7 v8 ) = 2n(m − 1) + 12, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 22, f (v11 v12 ) = 2n(m − 1) + 20, f (v12 v13 ) = 2n(m − 1) + 24, . . ., f (vn−9 vn−8 ) = 2mn − 18, f (vn−8 vn−7 ) = 2mn − 14, f (vn−7 vn−6 ) = 2mn − 16, f (vn−6 vn−5 ) = 2mn − 12, f (vn−5 vn−4 ) = 2mn − 10, f (vn−4 vn−3 ) = 2mn − 6, f (vn−3 vn−2 ) = 2mn − 8, f (vn−2 vn−1 ) = 2mn − 4, f (vn−1 vn ) = 2mn − 2, f (vn v1 ) = 2mn.
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Symmetry 2019, 11, 584
(2)
Label the edges of the circle Cn as follows: f (vn+1 vn+2 ) = 4n(m − 1) + 4, f (vn+2 vn+3 ) = 4n(m − 1) + 2, f (vn+3 vn+4 ) = 4n(m − 1) + 8, f (vn+4 vn+5 ) = 4n(m − 1) + 6, f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, 6 ≤ i ≤ n − 2, f (v2n−1 v2n ) = 2n(2m − 1), f (v2n vn+1 ) = 2n(2m − 1) − 2. Label the edges of (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
(4)
Cn , Cn , Cn , . . . , Cn and Cn , Cn , Cn , . . . , Cn as in case (1). Thus, the labels of corresponding vertices mod(4mn − 2n) will be: (1)
The label the vertices of Cn are: f ∗ (v1 ) ≡ 4, f ∗ (v2 ) ≡ 4mn − 4n + 12, f ∗ (v3 ) ≡ 4mn − 4n + 16, f ∗ (v4 ) ≡ 4mn − 4n + 20, f ∗ (v5 ) ≡ 4mn − 4n + 28, f ∗ (v6 ) ≡ 4mn − 4n + 36, f ∗ (v7 ) ≡ 4mn − 4n + 40, f ∗ (v8 ) ≡ 4mn − 4n + 44, f ∗ (v9 ) ≡ 4mn − 4n + 52, f ∗ (v10 ) ≡ 4mn − 4n + 60, f ∗ (v11 ) ≡ 4mn − 4n + 64, f ∗ (v12 ) ≡ 4mn − 4n + 68, f ∗ (v13 ) ≡ 4mn − 4n + 76, . . . , f ∗ (vn−8 ) ≡ 4n − 48, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 40, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 24, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 16, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n − 2. (2)
The label the vertices of the circle Cn are: f ∗ (vn+1 ) ≡ 6, f ∗ (vn+2 ) ≡ 10, f ∗ (vn+3 ) ≡ 14, f ∗ (vn+4 ) ≡ 18, ≡ 20, f ∗ (vn+i ) ≡ 4i + 2, 6 ≤ i ≤ n − 2, f ∗ (v2n−1 ) ≡ 4n, f ∗ (v2n ) ≡ 4n + 2. f ∗ (vn+5 )
(3)
(4)
(m−2)
The label the vertices of Cn , Cn , . . . , Cn
(m−1)
, Cn
(m)
, Cn
respectively are as the same as in case (1).
Remark 2. In case n = 2. Let the edges of the cylinder grid graph Cm,2 are labeled as shown in Figure 10. The corresponding labels of vertices mod(8m − 4) are as follows: f ∗ (v1 ) ≡ 8, f ∗ (v2i+1 ) ≡ 4m + 8i + 4, 1 ≤ m−1 ∗ ∗ ∗ ∗ i ≤ m−3 2 , f (v2i ) ≡ 8i + 6, 1 ≤ i ≤ 2 ; f (v 1 ) ≡ 12, f (v 2 ) ≡ 20, f (v 2i+1 ) ≡ 4m + 8i + 18, 1 ≤ i ≤ m−3 ∗ m−1 , f ( v ) ≡ 8i + 10, 2 ≤ i ≤ . 2i 2 2
Figure 10. The cylinder grid graph Cm,2 .
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Symmetry 2019, 11, 584
Case (3): n ≡ 4mod12. (k ) (1) First we label the edges of the paths Pm , 1 ≤ k ≤ n begin with the edges of the path Pm as the same in case (1). (k ) Second we label the edges of the circles Cn , 1 ≤ k ≤ m begin with the edges of the inner most (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , then the edges of the circles Cn Finally we label the edges of the circles (1)
(m−1) (m−3) (2) , Cm , . . . , Cm . Cm
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 6, f (v3 v4 ) = 2n(m − 1) + 4, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 14, f (v7 v8 ) = 2n(m − 1) + 12, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 22, f (v11 v12 ) = 2n(m − 1) + 20, f (v12 v13 ) = 2n(m − 1) + 24, . . . , f (vn−8 vn−7 ) = 2mn − 16, f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 2, f (vn−2 vn−1 ) = 2mn − 6, f (vn−1 vn ) = 2mn, and f (vn v1 ) = 2mn − 4. (m)
(m−2)
Label the edges of Cn , Cn
(m−4)
, Cn
(3)
(m−1)
, . . . , Cn and Cn
(m−3)
, Cn
(m−5)
, Cn
(4)
(2)
, . . . , Cn , Cn as in case (1).
(1) Cn
Thus we have the labels of corresponding vertices of the circle mod(4mn − 2n) will be: f ∗ (v1 ) ≡ 0, f ∗ (v2 ) ≡ 4mn − 4n + 12, f ∗ (v3 ) ≡ 4mn − 4n + 16, f ∗ (v4 ) ≡ 4mn − 4n + 20, f ∗ (v5 ) ≡ 4mn − 4n + 28, f ∗ (v6 ) ≡ 4mn − 4n + 36, f ∗ (v7 ) ≡ 4mn − 4n + 40, f ∗ (v8 ) ≡ 4mn − 4n + 44, f ∗ (v9 ) ≡ 4mn − 4n + 52, f ∗ (v10 ) ≡ 4mn − 4n + 60, f ∗ (v11 ) ≡ 4mn − 4n + 64, f ∗ (v12 ) ≡ 4mn − 4n + 68, f ∗ (v13 ) ≡ 4mn − 4n + 76, . . . , f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 36, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 28, f ∗ (vn−3 ) ≡ 4n − 16, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n − 4. (2)
(3)
(4)
(m−2)
The label the vertices of Cn , Cn , Cn , . . . , Cn
(m−1)
, Cn
(m)
, Cn
respectively are as same in case (1)
Remark 3. In case n = 4. Let the the edges of the cylinder grid graph Cm,4 are labeled as shown in Figure 11. The corresponding labels of vertices mod(16m − 8) are as follows: f ∗ (v1 ) ≡ 6, f ∗ (v2 ) ≡ 8, f ∗ (v3 ) ≡ 16, f ∗ (v4 ) ≡ 20; f ∗ (v4i+1 ) ≡ 4i + 10, 1 ≤ i ≤ 3, f ∗ (v8 ) ≡ 28; f ∗ (v8k+i ) ≡ 8m + 4i + 16k − 10, 1 ≤ i ≤ 4, 1 ≤ k ≤ m−5 ∗ ∗ ∗ ∗ ∗ 2 ; f (v4m−11 ) ≡ 0, f (v4m−10 ) ≡ 2, f (v4m−9 ) ≡ 4, f (v4m−8 ) ≡ 10, f (v8k+4+i ) ≡ 4i + 16k + 10, 1 ≤ i ≤ m−3 4, 1 ≤ k ≤ 2 . Case (4): n ≡ 6mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n begin with the edges of the path Pm as the same in case (1). (k ) Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ m begin with the edges of the inner most (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) , Cm , . . . , Cm . Cm
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 6, f (v3 v4 ) = 2n(m − 1) + 4, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 14, f (v7 v8 ) = 2n(m − 1) + 12, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 22, f (v11 v12 ) = 2n(m − 1) + 20, f (v12 v13 ) = 2n(m − 1) + 24, . . ., f (vn−9 vn−8 ) = 2mn − 18, f (vn−8 vn−7 ) = 2mn − 14, f (vn−7 vn−6 ) = 2mn − 16, f (vn−6 vn−5 ) = 2mn − 12, f (vn−5 vn−4 ) = 2mn − 10, f (vn−4 vn−3 ) = 2mn − 6, f (vn−3 vn−2 ) = 2mn − 8, f (vn−2 vn−1 ) = 2mn − 4, f (vn−1 vn ) = 2mn + 2, f (vn v1 ) = 2mn − 2. (m−4)
Label the edges of Cn
(3)
(m−1)
, . . . , Cn and Cn
(m−2) Cn
(m−3)
, Cn
(m−5)
, Cn
(4)
(2)
, . . . , Cn , Cn as in case (1).
Label the edges of the circle as follows: f (v(m−3)n+1 v(m−3)n+2 ) = 2n(m + 2), f (v(m−3)n+i v(m−3)n+i+1 ) = 2n(m + 1) + 2i, 2 ≤ i ≤ n − 1, f (v(m−2)n v(m−3)n+1 ) = 2n(m + 2) + 2. (m)
Label the edges of the circle Cn as follows: f (v(m−1)n+1 v(m−1)n+2 ) 2mn, f (v(m−1)n+i v(m−1)n+i+1 ) = 2mn + 2i, 2 ≤ i ≤ n − 1, f (v(m−2)n v(m−3)n+1 ) = 2n(m + 2).
53
=
Symmetry 2019, 11, 584
Figure 11. An e.e.g., l. of the cylinder grid graph Cm,4 .
Thus we have the labels of corresponding vertices mod(4mn − 2n) will be: (1)
The labels the vertices of the circle Cn are: f ∗ (v1 ) ≡ 2, f ∗ (v2 ) ≡ 4mn − 4n + 12, f ∗ (v3 ) ≡ 4mn − 4n + 16, f ∗ (v4 ) ≡ 4mn − 4n + 20, f ∗ (v5 ) ≡ 4mn − 4n + 28, f ∗ (v6 ) ≡ 4mn − 4n + 36, f ∗ (v7 ) ≡ 4mn − 4n + 40, f ∗ (v8 ) ≡ 4mn − 4n + 44, f ∗ (v9 ) ≡ 4mn − 4n + 52, f ∗ (v10 ) ≡ 4mn − 4n + 60, f ∗ (v11 ) ≡ 4mn − 4n + 64, f ∗ (v12 ) ≡ 4mn − 4n + 68, . . . , f ∗ (vn−8 ) ≡ 4n − 48, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 40, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 24, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 16, f ∗ (vn−1 ) ≡ 4n − 4; f ∗ (vn ) ≡ 4n. (m−2)
The labels the vertices of the circle Cn are: f ∗ (v(m−3)n+1 ) ≡ 4mn − 6n + 6; f ∗ (v(m−3)n+2 ) ≡ 4mn − 6n + 8; f ∗ (v(m−3)n+i ) ≡ 4mn − 6n + 4i + 2, 3 ≤ i ≤ n − 1, f ∗ (v(m−2)n ) ≡ 4mn − 4n + 4. (2)
(3)
(4)
(m−1)
The labels the vertices of Cn , Cn , Cn , . . . , Cn
(m) The labels the vertices of Cn
(m−3)
, Cn
respectively are the same as in case (1).
are: f ∗ (v(m−1)n+1 ) ≡ 2mn + 2, f ∗ (v(m−1)n+2 ) ≡ 2mn + 4, f ∗ (v(m−1)n+i ) ≡ 2mn + 2i + 2, 3 ≤ i ≤ n. Case (5): n ≡ 8mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n begin with the edges of the path Pm as the same in case (1). (k ) Secondly, we label the edges of the circles Cn , 1 ≤ k ≤ m begin with the edges of the inner most (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , then the edges of the circles Cn (m−1)
Finally we label the edges of the circles Cm (1) Cn
(m−3)
, Cm
(2)
(m−4)
, Cn
(3)
, . . . , Cn .
, . . . , Cm
Label the edges of the circle as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 6, f (v3 v4 ) = 2n(m − 1) + 4, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 14, f (v7 v8 ) = 2n(m − 1) + 12, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) =
54
Symmetry 2019, 11, 584
2n(m − 1) + 22, f (v11 v12 ) = 2n(m − 1) + 20, f (v12 v13 ) = 2n(m − 1) + 24, f (v13 v14 ) = 2n(m − 1) + 26, . . . , f (vn−9 vn−8 ) = 2mn − 20, f (vn−8 vn−7 ) = 2mn − 16, f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 6, f (vn−2 vn−1 ) = 2mn − 2, f (vn−1 vn ) = 2mn − 4, f (vn v1 ) = 2mn + 4. (m)
Label the edges of the circle Cn as follows f (v(m−1)n+1 v(m−1)n+2 ) = 2mn + 2, f (v(m−1)n+2 v(m−1)n+3 ) = 2mn + 6, f (v(m−1)n+i v(m−1)n+i+1 ) = 2mn + 2i + 2, 3 ≤ i ≤ n − 1, f (vmn v(m−1)n+1 ) = 2mn. (m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
(4)
(2)
Label the edges of Cn , Cn , . . . , Cn and Cn , Cn , Cn , . . . , Cn , Cn as the same in case (1). (1) Thus we labels of corresponding vertices of the circle Cn mod(4mn − 2n) will be: f ∗ (v1 ) ≡ ∗ ∗ ∗ 8, f (v2 ) ≡ 4mn − 4n + 12, f (v3 ) ≡ 4mn − 4n + 16, f (v4 ) ≡ 4mn − 4n + 20, f ∗ (v5 ) ≡ 4mn − 4n + 28, f ∗ (v6 ) ≡ 4mn − 4n + 36, f ∗ (v7 ) ≡ 4mn − 4n + 40, f ∗ (v8 ) ≡ 4mn − 4n + 44, f ∗ (v9 ) ≡ 4mn − 4n + 52, f ∗ (v10 ) ≡ 4mn − 4n + 60, f ∗ (v11 ) ≡ 4mn − 4n + 64, f ∗ (v12 ) ≡ 4mn − 4n + 68, f ∗ (v13 ) ≡ 4mn − 4n + 76, . . ., f ∗ (vn−8 ) ≡ 4n − 52, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 36, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 28, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n. (m)
The labels the vertices of the circle Cn are: f ∗ (v(m−1)n+1 ) ≡ 2mn − 4n + 14, f ∗ (v(m−1)n+2 ) ≡ 2mn + 8, f ∗ (v(m−1)n+i ) ≡ 2mn + 2i + 6, 3 ≤ i ≤ n − 1, f ∗ (vmn ) ≡ 2mn + 4. The labels the vertices of (2)
(3)
(4)
(m−2)
Cn , Cn , Cn , . . . , Cn
(m−1)
, Cn
respectively are as the same in case (1). (k )
Case (6): n ≡ 10mod12. First we label the edges of the paths Pm , 1 ≤ k ≤ n begin with the edges of the path
(1) Pm
as the same as in case (1).
(k )
Second we label the edges of the circles Cn , 1 ≤ k ≤ m begin with the edges of the inner most circle
(1) Cn ,
then the edges of outer circle
(m) Cn ,
Finally we label the edges of the circles (1)
(m−2)
then the edges of the circles Cn
(m−1) (m−3) (2) , Cm , . . . , Cm . Cm
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 6, f (v3 v4 ) = 2n(m − 1) + 4, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 14, f (v7 v8 ) = 2n(m − 1) + 12, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 22, f (v11 v12 ) = 2n(m − 1) + 20, f (v12 v13 ) = 2n(m − 1) + 24, f (v13 v14 ) = 2n(m − 1) + 26, . . . , f (vn−9 vn−8 ) = 2mn − 18, f (vn−8 vn−7 ) = 2mn − 14, f (vn−7 vn−6 ) = 2mn − 16, f (vn−6 vn−5 ) = 2mn − 12, f (vn−5 vn−4 ) = 2mn − 10, f (vn−4 vn−3 ) = 2mn − 6, f (vn−3 vn−2 ) = 2mn − 8, f (vn−2 vn−1 ) = 2mn − 4, f (vn−1 vn ) = 2mn − 2, f (vn v1 ) = 2mn. (2)
Label the edges of the circle Cn as follows: f (vn+1 vn+2 ) = 4n(m − 1) + 4, f (vn+2 vn+3 ) = 4n(m − 1) + 2, f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, 3 ≤ i ≤ n − 2, f (v2n−1 v2n ) = 2n(2m − 1), f (v2n v2n+1 ) = 2n(2m − 1) − 2. (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
(4)
Label the edges of Cn , Cn , Cn , . . . , Cn and Cn , Cn , Cn , . . . , Cn as in case (1). Thus we have the labels of corresponding vertices mod(4mn − 2n) will be: (1)
The labels the vertices of the circle Cn are as follows: f ∗ (v1 ) ≡ 4, f ∗ (v2 ) ≡ 4mn − 4n + 12, f ∗ (v3 ) ≡ 4mn − 4n + 16, f ∗ (v4 ) ≡ 4mn − 4n + 20, f ∗ (v5 ) ≡ 4mn − 4n + 28, f ∗ (v6 ) ≡ 4mn − 4n + 36, f ∗ (v7 ) ≡ 4mn − 4n + 40, f ∗ (v8 ) ≡ 4mn − 4n + 44, f ∗ (v9 ) ≡ 4mn − 4n + 52, f ∗ (v10 ) ≡ 4mn − 4n + 60, f ∗ (v11 ) ≡ 4mn − 4n + 64, f ∗ (v12 ) ≡ 4mn − 4n + 68, f ∗ (v13 ) ≡ 4mn − 4n + 76, f ∗ (v14 ) ≡ 4mn − 4n + 84, . . . , f ∗ (vn−8 ) ≡ 4n − 48, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 40, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 24, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 16, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n − 2. (2)
The labels the vertices of the circle Cn are as follows: f ∗ (vn+1 ) ≡ 6, f ∗ (vn+2 ) ≡ 10, f ∗ (vn+3 ) ≡ ∗ ∗ 12, n+i ) ≡ 4i + 2, 4 ≤ i ≤ n − 2, f (v2n−1 ) ≡ 4n, f (v2n ) ≡ 4n + 2. Label the vertices of f ∗ (v
(3)
(4)
(m−2)
(m−1)
(m)
Cn , Cn , . . . , Cn , Cn , Cn respectively are as the same as in case (1). Illustration: The edge even graceful labeling of the cylinder C9,2 , C9,4 , C7,10 , C7,12 , C7,14 C7,16 C7,18 and C7,20 are shown in Figure 12.
55
grid
graphs
Symmetry 2019, 11, 584
(a) &
(b) & Figure 12. Cont.
56
Symmetry 2019, 11, 584
(c) &
(d) & Figure 12. Cont. 57
Symmetry 2019, 11, 584
(e) &
(f) & Figure 12. Cont. 58
Symmetry 2019, 11, 584
(g) &
(h) & Figure 12. An e.e.g., l. of the cylinder grid graphs C9,2 , C9,4 , C7,10 , C7,12 , C7,14 , C7,16 , C7,18 , and C7,20 .
59
Symmetry 2019, 11, 584
Theorem 4. If m is an odd positive integer greater than 3 and n is an odd positive integer, n ≥ 3, then the cylinder grid graph Cm,n , is an edge even graceful graph. Proof. Using standard notation p = V (Cm,n ) = mn, q = E(Cm,n ) = 2mn − n, r = max(p, q) = 2mn − n, and f : E(Cm,n ) → {2, 4, 6, . . . , 4mn − 2n − 2}. Let the cylinder grid graph Cm,n be as in Figure 9. There are six cases: Case (1): n ≡ 1mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path Pm as follows: Move clockwise to label the edges v1 vn+1 , v2 vn+2 , v3 vn+3 , . . . , vn−1 v2n−1 , vn v2n by 2, 4, 6, . . . , 2n − 2, 2n, then move anticlockwise to label the edges vn+1 vn+2 , v2n v3n , v2n−1 v3n−1 , . . . , vn+3 v2n+3 , vn+2 v2n+2 by 2n + 2, 2n + 4, 2n + 6, . . . , 4n − 2, 4n, then move clockwise to label the edges v2n+1 v3n+1 , v2n+2 v3n+2 , v2n+3 v3n+3 , . . . , v3n−1 v4n−1 , v3n v4n by 4n + 2, 4n + 4, 4n + 6, . . . , 6n − 2, 6n, and so on. Finally, move anticlockwise to label the edges v(m−2)n+1 v(m−1)n+1 , v(m−1)n vmn , v(m−1)n−1 vmn−1 , . . ., v(m−2)n+3 v(m−1)n+3 , v(m−2)n+2 vm(n−1)+2 by 2n(m − 2) + 2, 2n(m − 2) + 4, 2n(m − 2) + 6, . . . , 2n(m − 1) − 2, 2n(m − 1). (k )
Second, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost circle
(1) Cn ,
then the edges of outer circle
(m) Cn ,
Finally, we label the edges of the circles (1)
(m−2)
and then the edges of the circles Cn
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 4, f (v3 v4 ) = 2n(m − 1) + 8, f (v4 v5 ) = 2n(m − 1) + 6, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 12, f (v7 v8 ) = 2n(m − 1) + 16, f (v8 v9 ) = 2n(m − 1) + 14, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 20, f (v11 v12 ) = 2n(m − 1) + 24, f (v12 v13 ) = 2n(m − 1) + 22, f (v13 v14 ) = 2n(m − 1) + 26, f (v14 v15 ) = 2n(m − 1) + 28, . . . , f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 6, f (vn−2 vn−1 ) = 2mn − 2, f (vn−1 vn ) = 2mn − 4, f (vn v1 ) = 2mn. (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
(4)
(2)
Then, label the edges of Cn , Cn , Cn , . . . , Cn and Cn , Cn , Cn , . . . , Cn , Cn as follows: (m) Label the edges of the circle Cn as follows: f (v(m−1)n+i v(m−1)n+i+1 ) = 2mn + 2i, f (vmn v(m−1)n+1 ) = 2n(m + 1), 1 ≤ i ≤ n − 1. (m−2)
Label the edges of the circle Cn as follows: 2i, f (v(m−2)n v(m−3)n+1 ) = 2n(m + 2), 1 ≤ i ≤ n − 1.
f (v(m−3)n+i v(m−3)n+i+1 ) = 2n(m + 1) +
(m−4)
Label the edges of the circle Cn as follows: f (v(m−5)n+i v(m−5)n+i+1 ) = 2n(m + 2) + 2i, f (v(m−4)n v(m−5)n+1 ) = 2n(m + 3), 1 ≤ i ≤ n − 1, and so on. (3)
Label the edges of the circle Cn as follows: f (v2n+i v2n+i+1 ) = 3n(m − 1) + 2i, f (v3n v2n+1 ) = n(3m − 1), 1 ≤ i ≤ n − 1, (m−1)
Label the edges of the circle Cn as follows: 2i, f (v(m−1)n v(m−1)n+1 ) = n(3m + 1), 1 ≤ i ≤ n − 1,
f (v(m−2)n+i v(m−2)n+i+1 ) = n(3m − 1) +
(m−3)
Label the edges of the circle Cn as follows: f (v(m−4)n+i v(m−4)n+i+1 ) = n(3m + 2) + 2i, f (v(m−3)n v(m−4)n+1 ) = 3n(m + 1), 1 ≤ i ≤ n − 1, . . . , and so on. (4)
Label the edges of the circle Cn as follows: f (v3n+i v3n+i+1 ) = 2n(2m − 3) + 2i, f (v4n v3n+1 ) = 4n(m − 1), 1 ≤ i ≤ n − 1, (2)
Label the edges of Cn as follows: f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, f (v2n v2n+1 ) = 2n(m − 1), 1 ≤ i ≤ n − 1, (1) Thus, the labels of corresponding vertices of the circle Cn mod(4mn − 2n) will be: f ∗ (v1 ) ≡ 4, f ∗ (v2 ) ≡ 4mn − 4n + 10, f ∗ (v3 ) ≡ 4mn − 4n + 18, f ∗ (v4 ) ≡ 4mn − 4n + 22, f ∗ (v5 ) ≡ 4mn − 4n + 26, f ∗ (v6 ) ≡ 4mn − 4n + 34, f ∗ (v7 ) ≡ 4mn − 4n + 42, f ∗ (v8 ) ≡ 4mn − 4n + 46, f ∗ (v9 ) ≡ 4mn − 4n + 50, f ∗ (v10 ) ≡ 4mn −
60
Symmetry 2019, 11, 584
4n + 58, f ∗ (v11 ) ≡ 4mn − 4n + 66, f ∗ (v12 ) ≡ 4mn − 4n + 70, f ∗ (v13 ) ≡ 4mn − 4n + 74, f ∗ (v14 ) ≡ 4mn − 4n + 82, . . . , f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 36, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 28, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n − 4. (2)
(3)
(4)
(m−2)
(m−1)
(m)
The labels of the vertices of Cn , Cn , Cn , . . . , Cn , Cn , Cn , respectively, are as follows: ≡ 4i + 2; f ∗ (v2n+i ) ≡ 2mn + 4n + 4i + 2; f ∗ (v3n+i ) ≡ 4n + 4i + 2; . . . ; f ∗ (v(m−3)n+i ) ≡ 4mn − 6n + 4i + 2; f ∗ (v(m−2)n+i ) ≡ 2mn − 6n + 4i + 2; f ∗ (v(m−1)n+i ) ≡ 2mn + 2i + 2, 1 ≤ i ≤ n. Case (2): n ≡ 3mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path Pm as the same in case (1). (k ) Second, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost f ∗ (vn+i )
(1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , and then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 4, f (v3 v4 ) = 2n(m − 1) + 8, f (v4 v5 ) = 2n(m − 1) + 6, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 12, f (v7 v8 ) = 2n(m − 1) + 16, f (v8 v9 ) = 2n(m − 1) + 14, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 20, f (v11 v12 ) = 2n(m − 1) + 24, f (v12 v13 ) = 2n(m − 1) + 22, f (v13 v14 ) = 2n(m − 1) + 26, . . ., f (vn−9 vn−8 ) = 2mn − 18, f (vn−8 vn−7 ) = 2mn − 14, f (vn−7 vn−6 ) = 2mn − 16, f (vn−6 vn−5 ) = 2mn − 12, f (vn−5 vn−4 ) = 2mn − 10, f (vn−4 vn−3 ) = 2mn − 6, f (vn−3 vn−2 ) = 2mn − 8, f (vn−2 vn−1 ) = 2mn, f (vn−1 vn ) = 2mn − 2, f (vn v1 ) = 2mn − 4. (2)
Label the edges of the circle Cn as follows: f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, 1 ≤ i ≤ n − 9, f (v2n−9 v2n−8 ) = 2n(2m − 1) − 18, f (v2n−8 v2n−7 ) = 2n(2m − 1) − 14, f (v2n−7 v2n−6 ) = 2n(2m − 1) − 16, f (v2n−6 v2n−5 ) = 2n(2m − 1) − 10, f (v2n−5 v2n−4 ) = 2n(2m − 1) − 12, f (v2n−4 v2n−3 ) = 2n(2m − 1) − 6, f (v2n−3 v2n−2 ) = 2n(2m − 1) − 8, f (v2n−2 v2n−1 ) = 2n(2m − 1) − 4, f (v2n−1 v2n ) = 2n(2m − 1) − 2, f (v2n vn+1 ) = 2n(2m − 1). (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
, Cn , . . . , Cn and Cn , Cn , Cn Label the edges of Cn , Cn Thus, the labels of corresponding vertices mod(4mn − 2n) will be:
(4)
, . . . , Cn as in case (1).
(1)
The labels of the vertices of Cn are as follows: f ∗ (v1 ) ≡ 0, f ∗ (v2 ) ≡ 4mn − 4n + 10, f ∗ (v3 ) ≡ 4mn − 4n + 18, f ∗ (v4 ) ≡ 4mn − 4n + 22, f ∗ (v5 ) ≡ 4mn − 4n + 26, f ∗ (v6 ) ≡ 4mn − 4n + 34, f ∗ (v7 ) ≡ 4mn − 4n + 42, f ∗ (v8 ) ≡ 4mn − 4n + 46, f ∗ (v9 ) ≡ 4mn − 4n + 50, f ∗ (v10 ) ≡ 4mn − 4n + 58, f ∗ (v11 ) ≡ 4mn − 4n + 66, f ∗ (v12 ) ≡ 4mn − 4n + 70, f ∗ (v13 ) ≡ 4mn − 4n + 74, f ∗ (v14 ) ≡ 4mn − 4n + 82, . . . , f ∗ (vn−8 ) ≡ 4n − 48, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 40, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 24, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 4, f ∗ (vn ) ≡ 4n − 6. (2)
The labels of the vertices of the circle Cn are as follows: f ∗ (vn+i ) ≡ 4i + 2, 1 ≤ i ≤ n − 9, ≡ 4n − 28, f ∗ (v2n−7 ) ≡ 4n − 26, f ∗ (v2n−6 ) ≡ 4n − 22, f ∗ (v2n−5 ) ≡ 4n − 18, f ∗ (v2n−4 ) ≡ 4n − 14, f ∗ (v2n−3 ) ≡ 4n − 10, f ∗ (v2n−2 ) ≡ 4n − 8, f ∗ (v2n−1 ) ≡ 4n − 2, f ∗ (v2n ) ≡ 4n + 2. f ∗ (v2n−8 )
(3)
(4)
(m−2)
The labels of the vertices of Cn , Cn , . . . , Cn
(m−1)
, Cn
(m)
, Cn , respectively, are the same as in case (1).
Remark 4. In case n = 3 and m is odd, m ≥ 3. Let the label of edges of the cylinder grid graph Cm,3 be as in Figure 13. Thus, the labels of corresponding vertices mod(12m − 6) are as follows: (1)
The labels of the vertices of the circle C3 are f ∗ (v1 ) ≡ 8, f ∗ (v2 ) ≡ 12, f ∗ (v3 ) ≡ 16. (3)
The labels of the vertices of the circle C3 are f ∗ (v3m−2 ) ≡ 6m + 10, f ∗ (v3m−1 ) ≡ 6m + 12, f ∗ (v3m ) ≡ 6m + 14. (2) (4) (m−1) The labels of the vertices of the circles C3 , C3 , . . . , C3 are f ∗ (v3k+i ) ≡ 4i + 6k + 4, 1 ≤ i ≤ 3, 1 ≤ k ≤ m − 2, k is odd. (3) (5) (m−2) The labels of the vertices of the circles C3 , C3 , . . . , C3 are f ∗ (v3k+i ) ≡ 6m + 4i + 6k + 10, 1 ≤ i ≤ 3, 2 ≤ k ≤ m − 3, k is even.
61
Symmetry 2019, 11, 584
Figure 13. The cylinder grid graph Cm,3 , m is odd, m ≥ 3.
Case (3): n ≡ 5mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path Pm the same as in case (1). (k ) Second, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , and then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 4, f (v3 v4 ) = 2n(m − 1) + 8, f (v4 v5 ) = 2n(m − 1) + 6, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 12, f (v7 v8 ) = 2n(m − 1) + 16, f (v8 v9 ) = 2n(m − 1) + 14, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 20, f (v11 v12 ) = 2n(m − 1) + 24, f (v12 v13 ) = 2n(m − 1) + 22, f (v13 v14 ) = 2n(m − 1) + 26, . . . , f (vn−8 vn−7 ) = 2mn − 16, f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 6, f (vn−2 vn−1 ) = 2mn − 2, f (vn−1 vn ) = 2mn − 4, (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
(4)
(2)
f (vn v1 ) = 2mn. Label the edges of Cn , Cn , Cn , . . . , Cn and Cn , Cn , Cn , . . . , Cn , Cn as in case (1). (1) Thus, the labels of corresponding vertices of the circle Cn mod(4mn − 2n) will be: f ∗ (v1 ) ≡ ∗ ∗ ∗ 4, f (v2 ) ≡ 4mn − 4n + 10, f (v3 ) ≡ 4mn − 4n + 18, f (v4 ) ≡ 4mn − 4n + 22, f ∗ (v5 ) ≡ 4mn − 4n + 26, f ∗ (v6 ) ≡ 4mn − 4n + 34, f ∗ (v7 ) ≡ 4mn − 4n + 42, f ∗ (v8 ) ≡ 4mn − 4n + 46, f ∗ (v9 ) ≡ 4mn − 4n + 50, f ∗ (v10 ) ≡ 4mn − 4n + 58, f ∗ (v11 ) ≡ 4mn − 4n + 66, f ∗ (v12 ) ≡ 4mn − 4n + 70, f ∗ (v13 ) ≡ 4mn − 4n + 74, f ∗ (v14 ) ≡ 4mn − 4n + 82, . . . , f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 36, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 28, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n − 4. 62
Symmetry 2019, 11, 584
(2)
(3)
(4)
(m−2)
(m−1)
(m)
The labels of the vertices of Cn , Cn , Cn , . . . , Cn , Cn , Cn , respectively, are the same as in case (1). Case (4): n ≡ 7mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path Pm the same as in case (1). (k ) Second, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , and then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 4, f (v3 v4 ) = 2n(m − 1) + 8, f (v4 v5 ) = 2n(m − 1) + 6, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 12, f (v7 v8 ) = 2n(m − 1) + 16, f (v8 v9 ) = 2n(m − 1) + 14, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 20, f (v11 v12 ) = 2n(m − 1) + 24, f (v12 v13 ) = 2n(m − 1) + 22, f (v13 v14 ) = 2n(m − 1) + 26, . . . , f (vn−10 vn−9 ) = 2mn − 20, f (vn−9 vn−8 ) = 2mn − 18, f (vn−8 vn−7 ) = 2mn − 14, f (vn−7 vn−6 ) = 2mn − 16, f (vn−6 vn−5 ) = 2mn − 12, f (vn−5 vn−4 ) = 2mn − 10, f (vn−4 vn−3 ) = 2mn − 6, f (vn−3 vn−2 ) = 2mn − 8, f (vn−2 vn−1 ) = 2mn, f (vn−1 vn ) = 2mn − 2, f (vn v1 ) = 2mn − 4. (2)
Label the edges of the circle Cn as follows: f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, 1 ≤ i ≤ n − 9, f (v2n−9 v2n−8 ) = 2n(2m − 1) − 18, f (v2n−8 v2n−7 ) = 2n(2m − 1) − 14, f (v2n−7 v2n−6 ) = 2n(2m − 1) − 16, f (v2n−6 v2n−5 ) = 2n(2m − 1) − 10, f (v2n−5 v2n−4 ) = 2n(2m − 1) − 12, f (v2n−4 v2n−3 ) = 2n(2m − 1) − 6, f (v2n−3 v2n−2 ) = 2n(2m − 1) − 8, f (v2n−2 v2n−1 ) = 2n(2m − 1) − 4, f (v2n−1 v2n ) = 2n(2m − 1) − 2, f (v2n vn+1 ) = 2n(2m − 1). (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
, Cn , . . . , Cn and Cn , Cn , Cn Label the edges of Cn , Cn Thus, the labels of corresponding vertices mod(4mn − 2n) will be:
(4)
, . . . , Cn as in case (1).
(1)
The labels of the vertices of the circle Cn are as follows: f ∗ (v1 ) ≡ 0, f ∗ (v2 ) ≡ 4mn − 4n + ∗ ∗ ∗ 10, 3 ) ≡ 4mn − 4n + 18, f (v4 ) ≡ 4mn − 4n + 22, f (v5 ) ≡ 4mn − 4n + 26, f (v6 ) ≡ 4mn − 4n + ∗ ∗ ∗ ∗ 34, f (v7 ) ≡ 4mn − 4n + 42, f (v8 ) ≡ 4mn − 4n + 46, f (v9 ) ≡ 4mn − 4n + 50, f (v10 ) ≡ 4mn − 4n + 58, f ∗ (v11 ) ≡ 4mn − 4n + 66, f ∗ (v12 ) ≡ 4mn − 4n + 70, f ∗ (v13 ) ≡ 4mn − 4n + 74, f ∗ (v14 ) ≡ 4mn − 4n + 82, . . . , f ∗ (vn−9 ) ≡ 4n − 56, f ∗ (vn−8 ) ≡ 4n − 48, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 40, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 24, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 4, f ∗ (vn ) ≡ 4n − 6. f ∗ (v
(2)
The labels of the vertices of the circle Cn are as follows: f ∗ (vn+i ) ≡ 4i + 2, 1 ≤ i ≤ n − 9, f ∗ (v2n−8 ) ≡ 4n − 28, f ∗ (v2n−7 ) ≡ 4n − 26, f ∗ (v2n−6 ) ≡ 4n − 22, f ∗ (v2n−5 ) ≡ 4n − 18, f ∗ (v2n−4 ) ≡ 4n − 14, f ∗ (v2n−3 ) ≡ 4n − 10, f ∗ (v2n−2 ) ≡ 4n − 8, f ∗ (v2n−1 ) ≡ 4n − 2, f ∗ (v2n ) ≡ 4n + 2. (3)
(4)
(m−2)
(m−1)
(m)
, Cn , Cn , respectively, are as in case (1). The labels of the vertices of Cn , Cn , . . . , Cn Case (5): n ≡ 9mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path Pm the same as in case (1). (k ) Second, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , and then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 2, f (v2 v3 ) = 2n(m − 1) + 4, f (v3 v4 ) = 2n(m − 1) + 8, f (v4 v5 ) = 2n(m − 1) + 6, f (v5 v6 ) = 2n(m − 1) + 10, f (v6 v7 ) = 2n(m − 1) + 12, f (v7 v8 ) = 2n(m − 1) + 16, f (v8 v9 ) = 2n(m − 1) + 14, f (v9 v10 ) = 2n(m − 1) + 18, f (v10 v11 ) = 2n(m − 1) + 20, f (v11 v12 ) = 2n(m − 1) + 24, f (v12 v13 ) = 2n(m − 1) + 22, f (v13 v14 ) = 2n(m − 1) + 26, . . . , f (vn−11 vn−10 ) = 2mn − 22, f (vn−10 vn−9 ) = 2mn − 18, f (vn−9 vn−8 ) = 2mn − 20, f (vn−8 vn−7 ) = 2mn − 16, f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 6, f (vn−2 vn−1 ) = 2mn, f (vn−1 vn ) = 2mn − 2, f (vn v1 ) = 2mn − 4. (2)
Label the edges of the circle Cn as follows: f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, 1 ≤ i ≤ n − 8, f (v2n−7 v2n−6 ) = 2n(2m − 1) − 12, f (v2n−6 v2n−5 ) = 2n(2m − 1) − 14, f (v2n−5 v2n−4 ) = 2n(2m − 1) − 6, f (v2n−4 v2n−3 ) = 2n(2m − 1) − 10, f (v2n−3 v2n−2 ) = 2n(2m − 1) − 8, f (v2n−2 v2n−1 ) = 2n(2m − 1) − 4, f (v2n vn+1 ) = 2n(2m − 1). 63
Symmetry 2019, 11, 584
(m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
, Cn , Cn , Cn , . . . , Cn and Cn Label the edges of Cn , Cn Thus, the labels of corresponding vertices mod(4mn − 2n) will be:
(4)
, . . . , Cn as in case (1).
(1)
The labels of the vertices of the circle Cn are as follows: f ∗ (v1 ) ≡ 0, f ∗ (v2 ) ≡ 4mn − 4n + ∗ ∗ ∗ 10, 3 ) ≡ 4mn − 4n + 18, f (v4 ) ≡ 4mn − 4n + 22, f (v5 ) ≡ 4mn − 4n + 26, f (v6 ) ≡ 4mn − 4n + 34, f ∗ (v7 ) ≡ 4mn − 4n + 42, f ∗ (v8 ) ≡ 4mn − 4n + 46, f ∗ (v9 ) ≡ 4mn − 4n + 50, f ∗ (v10 ) ≡ 4mn − 4n + 58, f ∗ (v11 ) ≡ 4mn − 4n + 66, f ∗ (v12 ) ≡ 4mn − 4n + 70, f ∗ (v13 ) ≡ 4mn − 4n + 74, f ∗ (v14 ) ≡ 4mn − 4n + 82, . . . , f ∗ (vn−10 ) ≡ 4n − 60, f ∗ (vn−9 ) ≡ 4n − 56, f ∗ (vn−8 ) ≡ 4n − 52, f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 36, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 28, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 10, f ∗ (vn−1 ) ≡ 4n − 4, f ∗ (vn ) ≡ 4n − 6. f ∗ (v
(2)
The labels of the vertices of the circle Cn are as follows: f ∗ (vn+i ) ≡ 4i + 2, 1 ≤ i ≤ n − 8, f ∗ (v2n−7 ) ≡ 4n − 24, f ∗ (v2n−6 ) ≡ 4n − 22, f ∗ (v2n−5 ) ≡ 4n − 16, f ∗ (v2n−4 ) ≡ 4n − 12, f ∗ (v2n−3 ) ≡ 4n − 14, f ∗ (v2n−2 ) ≡ 4n − 8, f ∗ (v2n−1 ) ≡ 4n − 2, f ∗ (v2n ) ≡ 4n + 2. (3)
(4)
(m−2)
(m−1)
(m)
The labels of the vertices of Cn , Cn , . . . , Cn , Cn , Cn , respectively, are as in case (1). Case (6): n ≡ 11mod12. (k ) (1) First, we label the edges of the paths Pm , 1 ≤ k ≤ n beginning with the edges of the path Pm the same as in case (1). (k ) Second, we label the edges of the circles Cn , 1 ≤ k ≤ m beginning with the edges of the innermost (1)
(m)
(m−2)
circle Cn , then the edges of outer circle Cn , and then the edges of the circles Cn Finally, we label the edges of the circles (1)
(m−1) (m−3) (2) Cm , Cm , . . . , Cm .
(m−4)
, Cn
(3)
, . . . , Cn .
Label the edges of the circle Cn as follows: f (v1 v2 ) = 2n(m − 1) + 4, f (v2 v3 ) = 2n(m − 1) + 2, f (v3 v4 ) = 2n(m − 1) + 6, f (v4 v5 ) = 2n(m − 1) + 8, f (v5 v6 ) = 2n(m − 1) + 12, f (v6 v7 ) = 2n(m − 1) + 10, f (v7 v8 ) = 2n(m − 1) + 14, f (v8 v9 ) = 2n(m − 1) + 16, f (v9 v10 ) = 2n(m − 1) + 20, f (v10 v11 ) = 2n(m − 1) + 18, f (v11 v12 ) = 2n(m − 1) + 22, f (v12 v13 ) = 2n(m − 1) + 24, f (v13 v14 ) = 2n(m − 1) + 28, . . . , f (vn−8 vn−7 ) = 2mn − 16, f (vn−7 vn−6 ) = 2mn − 14, f (vn−6 vn−5 ) = 2mn − 10, f (vn−5 vn−4 ) = 2mn − 12, f (vn−4 vn−3 ) = 2mn − 8, f (vn−3 vn−2 ) = 2mn − 6, f (vn−2 vn−1 ) = 2mn − 2, f (vn−1 vn ) = 2mn − 4, f (vn v1 ) = 2mn. (2)
Label the edges of the circle Cn as follows: f (vn+i vn+i+1 ) = 4n(m − 1) + 2i, 1 ≤ i ≤ n − 2, f (v2n−1 v2n ) = 4mn, f (v2n vn+1 ) = 4mn − 2. (m)
(m−2)
(m−4)
(3)
(m−1)
(m−3)
(m−5)
Label the edges of Cn , Cn , Cn , . . . , Cn and Cn , Cn , Cn Thus, the labels of corresponding vertices mod(4mn − 2n) will be:
(4)
, . . . , Cn as in case (1).
(1)
The labels of the vertices of the circle Cn are as follows: f ∗ (v1 ) ≡ 6, f ∗ (v2 ) ≡ 4mn − 4n + 10, f ∗ (v3 ) ≡ 4mn − 4n + 14, f ∗ (v4 ) ≡ 4mn − 4n + 22, f ∗ (v5 ) ≡ 4mn − 4n + 30, f ∗ (v6 ) ≡ 4mn − 4n + 34, f ∗ (v7 ) ≡ 4mn − 4n + 38, f ∗ (v8 ) ≡ 4mn − 4n + 46, f ∗ (v9 ) ≡ 4mn − 4n + 54, f ∗ (v10 ) ≡ 4mn − 4n + 58, f ∗ (v11 ) ≡ 4mn − 4n + 62, f ∗ (v12 ) ≡ 4mn − 4n + 70, . . . , f ∗ (vn−7 ) ≡ 4n − 44, f ∗ (vn−6 ) ≡ 4n − 36, f ∗ (vn−5 ) ≡ 4n − 32, f ∗ (vn−4 ) ≡ 4n − 28, f ∗ (vn−3 ) ≡ 4n − 20, f ∗ (vn−2 ) ≡ 4n − 12, f ∗ (vn−1 ) ≡ 4n − 8, f ∗ (vn ) ≡ 4n − 4. (2)
The labels of the vertices of the circle Cn are as follows: f ∗ (vn+1 ) ≡ 4, f ∗ (vn+i ) ≡ 4i + 2, 2 ≤ i ≤ n − 2, f ∗ (v2n−1 ) ≡ 4n, f ∗ (v2n ) ≡ 4n + 2. (3)
(4)
(m−2)
(m−1)
(m)
The labels of the vertices of Cn , Cn , . . . , Cn , Cn , Cn , respectively, are as the same as in case (1). Illustration: An e.e.g.l. of the cylinder grid graphs C9,3 , C7,9 , C7,11 , C7,13 , C7,15 , C7,17 and C7,19 is shown in Figure 14.
64
Symmetry 2019, 11, 584
(a) C
(b) C Figure 14. Cont.
65
Symmetry 2019, 11, 584
(c) C
(d) C Figure 14. Cont.
66
Symmetry 2019, 11, 584
(e) C
(f) C Figure 14. Cont.
67
Symmetry 2019, 11, 584
(g) C Figure 14. An e.e.g.l. of the cylinder grid graphs C9,3 , C7,9 , C7,11 , C7,13 , C7,15 , C7,17 and C7,19 .
3. Conclusions In this paper, using the connection of labeling of graphs with modular arithmetic and theory of numbers in general, we give a detailed study for e.e.g., l. of all cases of members of the cylinder grid graphs. The study of necessary and suﬃcient conditions for e.e.g., l. of other important families including torus Cm × Cn and rectangular Pm × Pn grid graphs should be taken into consideration in future studies of e.e.g., l. Author Contributions: All authors contributed equally to this work. Funding: This work was supported by the deanship of Scientiﬁc Research, Taibah University, AlMadinah AlMunawwarah, Saudi Arabia. Acknowledgments: The authors are grateful to the anonymous reviewers for their helpful comments and suggestions for improving the original version of the paper. Conﬂicts of Interest: The authors declare that there are no conﬂicts of interest regarding the publication of this paper.
References 1. 2.
3. 4.
Acharya, B.D.; Arumugam, S.; Rosa, A. Labeling of Discrete Structures and Applications; Narosa Publishing House: New Delhi, India, 2008; pp. 1–14. Bloom, G.S. Numbered Undirected Graphs and Their Uses, a Survey of a Unifying Scientiﬁc and Engineering Concept and Its Use in Developing a Theory of NonRedundant Homometric Sets Relating to Some Ambiguities in Xray Diﬀraction Analysis. Ph.D. Thesis, University of Southern California, Los Angeles, CA, USA, 1975. Bloom, G.S.; Golomb, S.W. Numbered complete graphs, unusual rulers, and assorted applications. In Theory and Applications of Graphs, Lecture Notes in Math, 642; Springer: New York, NY, USA, 1978; pp. 53–65. Bloom, G.S.; Golomb, S.W. Applications of numbered undirected graphs. Proc. IEEE 1977, 65, 562–570. [CrossRef]
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5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Bloom, G.S.; Hsu, D.F. On graceful digraphs and a problem in network addressing. Congr. Numer. 1982, 35, 91–103. Graham, R.L.; Pollak, H.O. On the addressing problem for loop switching. Bell Syst. Tech. J. 1971, 50, 2495–2519. [CrossRef] Sutton, M.; Labellings, S.G. Summable Graphs Labellings and Their Applications. Ph.D. Thesis, the University of Newcastle, New South Wales, Australia, 2001. Shang, Y. More on the normalized Laplacian Estrada index. Appl. Anal. Discret. Math. 2014, 8, 346–357. [CrossRef] Shang, Y. Geometric assortative growth model for smallworld networks. Sci. World J. 2014, 2014, 1–8. [CrossRef] [PubMed] Shang, Y. Deﬀuant model of opinion formation in onedimensional multiplex networks. J. Phys. A Math. Theor. 2015, 48, 395101. [CrossRef] Gross, J.; Yellen, J. Graph Theory and Its Applications; CRC Press: Boca Raton, FL, USA, 1999. Rosa, A. On certain valuations of the vertices of a graph. In Theory of Graphs, Proceedings of the International Symposium, Rome, Italy, July 1966; Gordan and Breach, Dunod: New York, NY, USA, 1967; pp. 349–355. Golomb, S.W. How to Number a Graph. In Graph Theory and Computing; Read, R.C., Ed.; Cademic Press: New York, NY, USA, 1972; pp. 23–37. Gnanajothi, R.B. Topics in Graph Theory. Ph.D. Thesis, Madurai Kamaraj University, Tamil Nadu, India, 1991. Seoud, M.A.; AbdelAal, M.E. On odd graceful graphs. Ars Comb. 2013, 108, 161–185. Gao, Z. Odd graceful labelings of some union graphs. J. Nat. Sci. Heilongjiang Univ. 2007, 24, 35–39. Lo, S.P. On edgegraceful1abelings of graphs. Congr. Numer. 1985, 50, 231–241. Kuan, Q.; Lee, S.; Mitchem, J.; Wang, A. On edgegraceful unicyclic graphs. Congr. Numer. 1988, 61, 65–74. Lee, L.; Lee, S.; Murty, G. On edgegraceful labelings of complete graphs: Solutions of Lo’s conjecture. Congr. Numer. 1988, 62, 225–233. Solairaju, A.; Chithra, K. Edgeodd graceful graphs. Electron. Notes Discret. Math. 2009, 33, 15–20. [CrossRef] Daoud, S.N. Edge odd graceful labeling of some path and cycle related graphs. AKCE Int. J. Graphs Comb. 2017, 14, 178–203. [CrossRef] Daoud, S.N. Edge odd graceful labeling cylinder grid and torus grid graphs. IEEE Access 2019, 7, 10568–10592. [CrossRef] Daoud, S.N. Vertex odd graceful labeling. Ars Comb. 2019, 142, 65–87. Elsonbaty, A.; Daoud, S.N. Edge even graceful labeling of some path and cycle related graphs. Ars Comb. 2017, 130, 79–96. Daoud, S.N. Edge even graceful labeling polar grid graph. Symmetry 2019, 11, 38. [CrossRef] Gallian, J.A. A dynamic survey of graph labeling. Electron. J. Comb. 2017, 22, #DS6. © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups Xiaohong Zhang * and Xiaoying Wu Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China; [email protected] * Correspondence: [email protected] Received: 12 March 2019; Accepted: 8 April 2019; Published: 17 April 2019
Abstract: In this paper, some basic properties and structure characterizations of AGgroups are further studied. First, some examples of inﬁnite AGgroups are given, and weak commutative, alternative and quasicancellative AGgroups are discussed. Second, two new concepts of involution AGgroup and generalized involution AGgroup are proposed, the relationships among (generalized) involution AGgroups, commutative groups and AGgroups are investigated, and the structure theorems of (generalized) involution AGgroups are proved. Third, the notion of ﬁlter of an AGgroup is introduced, the congruence relation is constructed from arbitrary ﬁlter, and the corresponding quotient structure and homomorphism theorems are established. Keywords: Abel–Grassmann’s groupoid (AGgroupoid); Abel–Grassmann’s group (AGgroup); involution AGgroup; commutative group; ﬁlter
1. Introduction Nowadays, the theories of groups and semigroups [1–5] are attracting increasing attention, which can be used to express various symmetries and generalized symmetries in the real world. Every group or semigroup has a binary operation that satisﬁes the associative law. On the other hand, nonassociative algebraic structures have great research value. Euclidean space R3 with multiplication given by the vector cross product is an example of an algebra that is not associative, at the same time; Jordan algebra and Lie algebra are nonassociative. For the generalization of commutative semigroup, the notion of an AGgroupoid (Abel–Grassmann’s groupoid) is introduced in [6], which is also said to be a left almost semigroup (LAsemigroup). Moreover, a class of nonassociative ring with condition x(yz) = z(yx) is investigated in [7]; in fact, the condition x(yz) = z(yx) is a dual distortion of the operation law in AGgroupoids. An AGgroupoid is a nonassociative algebraic structure, but it is a groupoid (N, *) satisfying the left invertive law: (a ∗ b) ∗ c = (c ∗ b) ∗ a, for any a, b, c ∈ N. Now, many characterizations of AGgroupoids and various special subclasses are investigated in [8–13]. As a generalization of commutative group (Abelian group) and a special case of quasigroup, Kamran extended the concept of AGgroupoid to AGgroup in [14]. An AGgroupoid is called AGgroup if there exists left identity and inverse, and its many properties (similar to the properties of groups) have been revealed successively in [15,16]. In this paper, we further analyze and study the structural characteristics of AGgroups, reveal the relationship between AGgroups and commutative groups, and establish ﬁlter and quotient algebra theories of AGgroups. The paper is organized as follows. Section 2 presents several basic concepts and results. Some new properties of AGgroups are investigated in Section 3, especially some examples of inﬁnite AGgroups, and the authors prove that every weak commutative or alternative AGgroup is Symmetry 2019, 11, 553; doi:10.3390/sym11040553
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a commutative group (Abelian group) and every AGgroup is quasicancellative. In Section 4, two special classes of AGgroups are studied and the structure theorems are proved. In Section 5, the ﬁlter theory of AGgroups is established, the quotient structures induced by ﬁlters are constructed, and some homomorphism theorems are proved. Finally, the main results of this paper are systematically summarized via a schematic ﬁgure. 2. Preliminaries First, we present some basic notions and properties. A groupoid (N, *) is called an AGgroupoid (Abel–Grassmann’s groupoid), if for any a, b, c∈N, (a*b)*c = (c*b)*a. It is easy to verify that in an AGgroupoid (N, *), the medial law holds:
(a ∗ b) ∗ (c ∗ d) = (a ∗ c) ∗ (b ∗ d), for any a, b, c, d ∈ N. Let (N, *) be an AGgroupoid with left identity e, we have a ∗ (b ∗ c) = b ∗ (a ∗ c), for any a, b, c ∈ N;
(a ∗ b) ∗ (c ∗ d) = (d ∗ b) ∗ (c ∗ a), for any a, b, c, d ∈ N. NN = N, N*e = N = e*N. An AGgroupoid (N, *) is called a locally associative AGgroupoid, if it satisﬁes a*(a*a) = (a*a)*a, ∀a∈N. An AGgroupoid (N, *) is called an AGband, if it satisﬁes a*a = a (∀a∈N). Deﬁnition 1. ([9,10]) Let (N, *) be an AGgroupoid. Then, N is called to be quasicancellative if for any a, b∈N, a = a*b and b2 = b*a imply that a = b; and
(1)
a = b*a and b2 = a*b imply that a = b.
(2)
Proposition 1. ([9,10]) Every AGband is quasicancellative. Deﬁnition 2. ([14,15]) An AGgroupoid (N, *) is called an AGgroup or a left almost group (LAgroup), if there exists left identity e∈N (that is e*a = a, for all a∈N), and there exists a−1 ∈N such that a−1 *a = a* a−1 = e (∀a∈N). Proposition 2. ([15]) Assume that (N, *) is an AGgroup. We get that (N, *) is a commutative Abel–Grassmann’s Group if and only if it is an associative AGGroup. Proposition 3. ([15]) Let (N, *) be an AGgroup with right identity e. Then, (N, *) is an Abelian group. Proposition 4. ([15]) Let (N, *) be an AGgroup. Then, (N, *) has exactly one idempotent element, which is the left identity. Proposition 5. ([11]) Let (N, *) be an AGgroupoid with a left identity e. Then, the following conditions are equivalent, (1) (2) (3) (4)
N is an AGgroup. Every element of N has a right inverse. Every element a of N has a unique inverse a−1 . The equation x*a = b has a unique solution for all a, b∈N.
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Proposition 6. ([16]) Let (N, *) be an AGgroup. Deﬁne a binary operation ◦ as follows: x◦y = (x*e)*y, for any x, y∈N. Then, (N, ◦) is an Abelian group, denote it by ret(N, *) = (N, ◦). 3. Some Examples and New Results of AGGroups In this section, we give some examples of AGgroups (including some inﬁnite examples), and investigate the characterizations of weak commutative AGgroups, alternative AGgroups and quasicancellative AGgroups. Moreover, we obtain two subalgebras from arbitrary AGgroup. Example 1. Let us consider the rotation transformations of a square. A square is rotated 90◦ , 180◦ and 270◦ to the right (clockwise) and they are denoted by ϕa , ϕb and ϕc , respectively (see Figure 1). There is of course the movement that does nothing, which is denoted by ϕe . The following ﬁgure gives an intuitive description of these transformations. Denote N = {ϕe , ϕa , ϕb , ϕc }.
Me
Ma
Mb
Mc
Figure 1. The rotation transformations of a square.
Obviously, two consecutive rotations have the following results: ϕe ϕe = ϕe , ϕa ϕc = ϕc ϕa = ϕe , ϕb ϕb = ϕe . That is, ϕe −1 = ϕe , ϕa −1 = ϕc , ϕb −1 = ϕb ,ϕc −1 = ϕa . Now, we deﬁne operations * on N as follows: ϕx *ϕy = ϕx −1 ϕy , ∀x, y∈{e, a, b, c}. Then, (N, *) satisﬁes the left invertive law, and the operation * is as follows in Table 1. We can verify that (N, *) is an AGGroup. Table 1. AGgroup generated by rotation transformations of a square. *
ϕe
ϕa
ϕb
ϕc
ϕe ϕa ϕb ϕc
ϕe ϕc ϕb ϕa
ϕa ϕe ϕc ϕb
ϕb ϕa ϕe ϕc
ϕc ϕb ϕa ϕe
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Example 2. Let X = {(a, b)a, b∈R−{0}}, where R represents the set of all real numbers. Deﬁne binary operation * as follows: (a, b) * (c, d) = (ac, d/b), for any (a, b), (c, d)∈ X. Then, [(a, b) * (c, d)] * (e, f) = (ac, d/b) * (e, f) = (ace, fb/d); [(e, f) * (c, d)] * (a, b) = (ec, d/f) * (a, b) = (ace, fb/d). Therefore, [(a, b) * (c, d)] * (e, f) = [(e, f) * (c, d)] * (a, b), that is, the operation * satisﬁes left invertive law. For any (a, b)∈X, (1, 1) is the left identity of (a, b) and (1/a, b) is the left inverse of (a, b): (1,1) * (a, b) = (a, b); (1/a, b) * (a, b) = (1, 1). Therefore, (X, *) is an AGGroup. Example 3. Let Y = {(a, b)a∈R, b = 1 or −1}, where R represents the set of all real numbers. Deﬁne binary operation * as follows: (a, b) * (c, d) = (ac, b/d), for any (a, b), (c, d)∈Y. Then, [(a, b) * (c, d)] * (e, f) = (ac, b/d) * (e, f) = (ace, b/df); [(e, f) * (c, d)] * (a, b) = (ec, f/d) * (a, b) = (ace, f/bd). Because b, f∈ {1, −1}, b2 = f 2 , and b/f = f/b. We can get b/df = f/bd. Therefore, [(a, b) * (c, d)] * (e, f) = [(e, f) * (c, d)] * (a, b), that is, the operation * satisﬁes left invertive law. Moreover, we can verify that (1, 1) is the left identity and (1/a, ±1) is the left inverse of (a, ±1), since (1, 1) * (a, b) = (a,1/b) = (a, b); (because b=1 or −1) (1/a, 1) * (a, 1) = (1, 1) and (1/a, −1) * (a, −1) = (1, 1). Therefore, (Y, *) is an AGgroup. Example 4. Let Z = {(a, b)a∈R, b = 1, −1, i, or −i}, where R represents the set of all real numbers and I represents the imaginary unit. Deﬁne binary operation * as follows: (a, b) * (c, d) = (ac, b/d), for any (a, b), (c, d) ∈Z Then, [(a, b) * (c, d)] * (e, f) = (ac, b/d) * (e, f) = (ace, b/df); [(e, f) * (c, d)] * (a, b) = (ec, f/d) * (a, b) = (ace, f/bd). Because b, f∈{1, −1, i, −i}, hence b2 = f 2 , and b/f = f/b. We can get b/df = f/bd. Therefore, [(a, b) * (c, d)] * (e, f) = [(e, f) * (c, d)] * (a, b), that is, the operation * satisﬁes left invertive law. Therefore, (Z, *) is an AGgroupoid. However, it is not an AGgroup, since (1, 1) * (a, 1) = (a, 1), (1, 1) * (a, −1) = (a, −1); (1, −1) * (a, i) = (a, i), (1, −1) * (a, −i) = (a, −i). That is, (1, 1) and (1, −1) are locally identity, not an identity. Deﬁnition 3. Assume that (N, *) is an AGgroup. (N, *) is said to be a weak commutative Abel– Grassmann’s group (AGgroup), if one of the following conditions holds:
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(1) (2) (3)
e*x−1 = x−1 *e, for all x in N; e*x = x*e, for all x in N; or x−1 *y−1 = y−1 *x−1 , for all x, y in N.
Theorem 1. Let (N, *) be an AGgroup. We can get that N is a weak commutative AGgroup if and only if it is an Abelian group. Proof. First, we prove that the Conditions (1)–(3) in Deﬁnition 3 are equivalent for an AGgroup (N, *). (1)→(2): Suppose thatCondition (1) holds in the AGgroup (N, *). For all x in N, by (x−1 ) −1 e*(x−1 )
−1
−1 (x−1 ) *e,
= x,
we have = that is, e*x = x*e. (2)→(3): Suppose that Condition (2) holds in the AGgroup (N, *). For all x, y in N, by Proposition 3, we know that N is an Abelian group, that is, x*y = y*x, it follows that x−1 *y−1 = y−1 *x−1 . (3)→(1): Suppose that Condition (3) holds in the AGgroup (N, *). Then, for all x in N, we have −1
−1
(e−1 ) *x−1 = x−1 * (e−1 ) , that is, e*x−1 = x−1 *e. Now, we prove that an AGgroup (N, *) satisfying Condition (2) in Deﬁnition 3 is an Abelian group. Through Condition (2), e*a = a*e for any a∈N. Then, a*e = e*a = a, which means that e is right identity. Applying Proposition 3, we get that (N, ∗) is an Abelian group. Moreover, obviously, every Abelian group is a weak commutative AGgroup. Therefore, the proof is completed. Theorem 2. Assume that (N, *) is an AGgroup, we have that (N, *) is quasicancellative AGgroupoid, that is, if it satisﬁes the following conditions, for any x, y∈N, (1) (2)
x = x * y and y2 = y*x imply that x = y; and x = y * x and y2 = x * y imply that x = y.
Proof. (1) Suppose that x = x*y and y2 = y*x, where x, y∈N. Then, x = x ∗ y = (e ∗ x) ∗ y = ( y ∗ x) ∗ e = y2 ∗e = (e ∗ y) ∗ y = y2 .
(a)
That is, x = y2 ; it follows that x*y = y*x. Moreover, we have y ∗ e = y ∗ (x−1 ∗x) = (e ∗ y) ∗ ( x−1 ∗x) = (e ∗ x−1 ) ∗ ( y ∗ x) = x−1 ∗( y ∗ x) = x−1 ∗ x = e.
(b)
x ∗ e = (x ∗ y) ∗ e = (x ∗ y) ∗ ( x−1 ∗x) = (x ∗ x−1 ) ∗ ( y ∗ x) = (x ∗ x−1 ) ∗ y2 = (x ∗ x−1 ) ∗ ( y ∗ y) = (x ∗ y) ∗ ( x−1 ∗y) = x ∗ (x−1 ∗y) = (e ∗ x) ∗ ( x−1 ∗y) = (e ∗ x−1 ) ∗ (x ∗ y) = x−1 ∗(x ∗ y) = x−1 ∗ x = e.
(c)
Combining Equations (b) and (c), we can get x = e*x = (y*e)*x = (x*e)*y = e*y = y. (2) Suppose that x=y*x and y2 =x*y, where x, y∈N. Then, x = y ∗ x = (e ∗ y) ∗ (x ∗ y) ∗ e = y2 ∗e = (e ∗ y) ∗ y = y2 ∗e = (e ∗ y) ∗ y = y2 .
(d)
That is, x = y2 ; it follows that x*y = y*x. Then, we have y ∗ e = y ∗ (x−1 ∗x) = (e ∗ y) ∗ (x−1 ∗x) = (e ∗ x−1 ) ∗ ( y ∗ x) = (e ∗ x−1 ) ∗ x = e. x ∗ e = x ∗ ( y−1 ∗y) = (e ∗ x) ∗ ( y−1 ∗y) = y−1 ∗(x ∗ y) =
y−1 ∗y2 = y−1 ∗( y ∗ y) = y ∗ ( y−1 ∗ y) = y ∗ e = e.
Combining Equations (e) and (f), we can get x = e*x = (y*e)*x = (x*e)*y = e*y = y. 74
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Hence, (N, *) is quasicancellative AGgroupoid. Deﬁnition 4. Let (N, *) be an AGgroup. Then, (N, *) is called to be alternative, if it satisﬁes one of the following conditions, a*(a*b) = (a*a)*b, ∀a, b∈N; or a*(b*b) = (a*b)*b, ∀a, b∈N.
(1) (2)
Theorem 3. Let (N, *) be an AGgroup. Then, (N, *) is alternative if and only if it is an Abelian group. Proof. (1) Suppose that (N, *) is an alternative AGgroup, then Condition (2) in Deﬁnition 4 holds. Then, for any a, b∈N, a*(b*b) = (a*b)*b. Putting b = e and applying left invertive law, we get that a*e = a*(e*e) = (a*e)*e = (e*e)*a = e*a = a; by Proposition 3, we know that (N, *) is an Abelian group. (2) Suppose that (N, *) is an alternative AGgroup, then Condition (1) in Deﬁnition 4 holds. For any a, b∈N, it satisﬁes a*(a*b) = (a*a)*b. Putting b = e, we have (a*a)*e = a*(a*e). According to the arbitrariness of a, we can get that ((a*e)*(a*e))*e = (a*e)*((a*e)*e). Then, a*a = (e*a)*a = (a*a)*e = ((a*a)*(e*e))*e = ((a*e)*(a*e))*e = (a*e)*((a*e)*e) = (a*e)*a. Let b*a = e, using Condition (1) in Deﬁnition 4, (a*a)*b = a*(a*b). It follows that (a*a)*b = ((a*e)*a)*b. Thus, a = e*a = (b*a)*a = (a*a)*b = ((a*e)*a)*b = (b*a)*(a*e) = e*(a*e) = a*e. Applying Proposition (3), we know that (N, *) is an Abelian group. Conversely, it is obvious that every Abelian group is an alternative AGgroup. Therefore, the proof is completed. Theorem 4. Let (N, *) be an AGgroup. Denote U(N) = {x∈N x = x*e}. Then, (1) (2)
U(N) is subalgebra of N. U(N) is maximal subgroup of N with identity e.
Proof. (1) Obviously, e ∈ U(N), that is, U(N) is not empty. Suppose x, y∈U(N), then x*e = x and y*e = y. Thus, x*y = (x*e)*(y*e) = (x*y)*e ∈ U(N). This means that U(N) is a subalgebra of N. (2) For any x∈U(N), that is, x*e = x. Assume that y is the left inverse of x in N, then y*x = e. Thus, x*y = (e*x)*y = ((y*x)*x)*y = (y*x)*(y*x) = e*e = e, y = e*y = (x*y)*y = ((x*e)*y)*y = ((y*e)*x)*y = (y*x)*(y*e) = e*(y*e) = y*e. It follows that y∈U(N). Therefore, U(N) is a group, and it is a subgroup of N with identity e. If M is a subgroup of N with identity e, and U(N)⊆M, then M is an Abelian group (by Proposition (3)) and satisﬁes x*e = e*x = x, for any x∈M. Thus, M⊆U(N), it follows that M = U(N). Therefore, U(N) is maximal subgroup of N with identity e.
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Theorem 5. Let (N, *) be an AGgroup. Denote P(N) = {x∈N∃a∈N, s.t x = a*a}. Then (1) (2)
P(N) is the subalgebra of N; f is a homomorphism mapping from N to P(N), where f: N→P(N), f(x)=x*x∈P(N).
Proof. (1) Obviously, e ∈ P(N), that is, P(N) is not empty. Suppose x, y∈P(N) and a, b∈N. Then, a*a = x and b*b = y. Thus, x*y = (a*a)*(b*b) = (a*b)*(a*b) ∈ P(N). This means that P(N) is a subalgebra of N. (2) For any x, y ∈N, we have f (x*y) = (x*y)*(x*y) = (x*x)*(y*y) = f (x)*f (y). Therefore, f is a homomorphism mapping from N to P(N). 4. Involution AGGroups and Generalized Involution AGGroups In this section, we discuss two special classes of AGgroups, that is, involution AGgroups and generalized involution AGgroups. Some research into the involutivity in AGgroupoids is presented in [16,17] as the foundation, and further results are given in this section, especially the close relationship between these algebraic structures and commutative groups (Abelian groups), and their structural characteristics. Deﬁnition 5. Let (N, *) be an AGgroup. If (N, *) satisﬁes a*a = e, for any a∈N, then (N, *) is called an involution AGGroup. We can verify that (N, *) in Example 1 is an involution AGGroup. Example 5. Denote N = {a, b, c, d}, deﬁne operations * on N as shown in Table 1. We can verify that (N, *) is an involution AGgroup (Table 2). Table 2. Involution AGgroup (N, *). *
a
b
c
d
a b c d
a b d c
b a c d
c d a b
d c b a
Example 6. Let (G, +) be an Abelian group. Deﬁne operations * on G as follows: x*y = (−x) + y, ∀x, y∈G where (−x) is the inverse of x in G. Then, (G, *) is an involution AGgroup. Denote (G, *) by der (G, +) (see [15]), and call it derived AGgroup by Abelian group (G, +). Theorem 6. Let (N, *) be an AGgroup. Then, (N, *) is an involution AGGroup if and only if it satisﬁes one of the following conditions: (1) (2)
P(N) = {e}, where P(N) is deﬁned as Theorem 5. (x*x)*x = x for any x∈N.
Proof. Obviously, (N, *) is an involution AGgroup if and only if P(N) = {e}. If (N, *) is an involution AGgroup, then apply Deﬁnition 5, for any x∈N, (x*x)*x = e*x = x.
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Conversely, if (N, *) satisﬁes the Condition (2), then for any x∈N, (x*x)*(x*x) = ((x*x)*x)*x = x*x. This means that (x*x) is an idempotent element. Using Proposition 4, we have x*x = e. Thus, (N, *) is an involution AGgroup. Theorem 7. Let (N, *) be an involution AGgroup. Then, (N, ◦) = ret (N, *) deﬁned in Proposition 6 is an Abelian group, and the derived AGgroup der (N, ◦) by ret (N, *) (see Example 5) is equal to (N, *), that is, der(ret(N, *)) = (N, *). Proof. (1) By Proposition 6 and Deﬁnition 5, ∀x, y, z∈N, we can get that x◦y = y◦x; x◦e = e◦x = x; (x◦y)◦z = x◦(y◦z); x◦x−1 = x−1 ◦x = e. This means that (N, ◦) = ret(N, *) is an Abelian group. (2) For any x, y∈der(ret(N, *)) = der(N, ◦) = (N, •), x•y = (−x)◦y = ((−x)∗e)*y = ((x∗e)∗e)*y = ((e∗e)∗x)*y = (e∗x)*y = x*y. That is, der(ret(N, *)) = (N, •)= (N, *). Deﬁnition 6. Let (N, *) be an AGgroup. Then, (N, *) is called a generalized involution AGgroup if it satisﬁes: for any x∈N, (x*x)*(x*x) = e. Obviously, every involution AGgroup is a generalized involution AGgroup. The inverse is not true, see the following example. Example 7. Denote N = {e, a, b, c}, and deﬁne the operations * on N as shown in Table 3. We can verify that (N, *) is a generalized involution AGgroup, but it is not an involution AGgroup. Table 3. Generalized involution AGgroup (N, *). *
e
a
b
c
e a b c
e a c b
a e b c
b c a e
c b e a
Theorem 8. Let (N, *) be a generalized involution AGgroup. Deﬁne binary relation ≈ on N as follows: x ≈ y ⇔ x ∗ x = y ∗ y, f or any x, y ∈ N. Then, (1) (2) (3) (4)
≈ is an equvalent relation on N, and we denote the equivalent class contained x by[x]≈ . The equivalent class contained e by [e]≈ is an involution subAGgroup. For any x, y, z∈N, x≈y implies x*z≈y*z and z*x≈z*y. The quotient (N/ ≈ , *) is an involution AGgroup.
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Proof. (1) For any a∈N, we have a*a = a*a, thus a≈a. If a≈b, then a*a = b*b; it is obvious that b≈a. If a≈b and b≈c, then a*a = b*b and b*b = c*c; it is obvious that a*a = c*c, that is, a≈c. Therefore, ≈ is an equivalent relation on N. (2) ∀x, y∈ [e]≈ , we have x*x = y*y = e*e = e, thus (x*y)*(x*y) = (x*x)*(y*y) = e*e = e. This means that [e]≈ is a subalgebra of N. Thus, [e]≈ is an involution subAGgroup of N. (3) Assume that x≈y, then x*x = y*y. Thus, (x*z)*(x*z) = (x*x)*(z*z) = (y*y)*(z*z) = (y*z)*(y*z); (z*x)*(z*x) = (z*z)*(x*x) = (z*z)*(y*y) = (z*y)*(z*y). It follows that x*z≈y*z and z*x≈z*y. (4) By (3), we know that (N/≈ , *) is an AGgroup. Moreover, for any x ∈ [a]≈ ∗[a]≈ = [a∗a]≈ , x ∗ x = (a ∗ a) ∗ (a ∗ a) By Deﬁnition 6, (a*a)*(a*a) = e. Then, x ∗ x = e for any x ∈ [a∗a]≈ . From this, we have x∈[e]≈ , [a*a]≈ ⊆ [e]≈ . Hence, [a*a]≈ = [e]≈ . That is, [a]≈ *[a]≈ = [e]≈ . Therefore, (N/≈ , *) is an involution AGgroup. Theorem 9. Let (N, *) be an AGgroup, denote I(N) = {x∈Nx*x=e}, GI(N) = {x∈N(x*x)*(x*x)=e}. Then, I(N) and GI(N) are subalgebra of N. I(N) is an involution AGgroup and GI(N) is a generalized involution AGgroup. Proof. (1) It is obvious that e∈I(N). For any x, y∈I(X), we have x*x = e and y*y = e. By medial law, (x*y)*(x*y) = (x*x)*(y*y) = e*e = e. Hence, I(N) is a subalgebra of N and I(N) is an involution AGgroup. (2) Obviously, e∈GI(N). Assume that x, y∈GI(X), then (x*x)*(x*x) = (y*y)*(y*y) = e. Thus, ((x*y)*(x*y))*((x*y)*(x*y)) = ((x*x)*(y*y))*((x*x)*(y*y)) = ((x*x)*(x*x))*((y*y)*(y*y)) = e*e = e. It follows that x*y∈GI(N), and GI(N) is a subalgebra of N. Moreover, from ((x*x)*x)*x = (x*x)*(x*x) = e, we get that a = (x*x)*x is the left inverse of x, and (a*a)*(a*a) = (((x*x)*x)*((x*x)*x))*(a*a) = ((x*x)*(x*x))*(x*x))*(a*a) = (e*(x*x))*(a*a) = (x*x)*(a*a) = (x*x)*(x*x) = e.
That is, a = (x*x)*x∈GI(N). It follows that GI(N) is an AGgroup. By the deﬁnition of GI(N), we get that GI(N) is a generalized involution AGgroup.
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5. Filter of AGGroups and Homomorphism Theorems Deﬁnition 7. Let (N, *) be an AGgroup. A nonempty subset F of N is called a ﬁlter of N if, for all x, y∈N, F satisﬁes the following properties, (1) (2) (3)
e∈F; x*x∈F; and x∈F and x*y∈F imply that y∈F.
If F is a ﬁlter and subalgebra of N, then F will be called a closed ﬁlter of N. Theorem 10. Let (N, *) be a generalized involution AGgroup, I(N) = {x∈N e=x*x} be the involution part of N (see Theorem 9). Then, I(N) is a closed ﬁlter of N. Proof. It is obvious that e∈I(N). ∀x∈N, since (x*x)*(x*x) = e, then x*x∈I(N). Moreover, assuming that x∈I(N) and x*y∈I(N), then e = x*x, (x*y)*(x*y) = e. Thus, y*y = e*(y*y) = (x*x)*(y*y) = (x*y)*(x*y) = e. Hence, y∈I(N), and I(N) is a ﬁlter of N. By Theorem 9, I(N) is a subalgebra of N. Therefore, I(N) is a closed ﬁlter of N. Theorem 11. Let (N, *) be an AGgroup and F be a closed ﬁlter of N. Deﬁne binary relation ≈F on N as follows: x ≈F y ⇔ (x ∗ y ∈ F, y ∗ x ∈ F), f or any x, y in N. Then, (1) (2) (3)
≈F is an equivalent relation on N. x≈F y and a≈F b imply x*a≈F y*b. f: N→N/F is a homomorphism mapping, where N/F = {[x]F : x∈N}, [x]F denote the equivalent class contained x.
Proof. (1) ∀x∈N, by Deﬁnition 7(2), x*x∈F. Thus, x≈F x. Assume x≈F y, then x*y∈F, y*x∈F. It follows that y≈F x. Suppose that x≈F y and y≈F z. We have x*y∈F, y*x∈F, y*z∈F and z*y∈F. By medial law and Deﬁnition 7, ( y ∗ y) ∗ (z ∗ x) = ( y ∗ z) ∗ ( y ∗ x) ∈F, then (z ∗ x) ∈ F;
( y ∗ y) ∗ (x ∗ z) = ( y ∗ x) ∗ ( y ∗ z) ∈F, then (x ∗ z) ∈ F. It follows that x≈F z. Therefore, ≈F is an equivalent relation on N. (2) Suppose that x≈F y and a≈F b. We have x*y∈F, y*x∈F, a*b∈F and b*a∈F. By medial law and Deﬁnition 7,
(x ∗ a) ∗ ( y ∗ b) = (x ∗ y) ∗ (a ∗ b) ∈F; ( y ∗ b) ∗ (x ∗ a) = ( y ∗ x) ∗ (b ∗ a) ∈ F. It follows that x*a≈F y*b. 79
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(3) Combining (1) and (2), we can obtain (3). The proof complete. Theorem 12. Let (N, *) be a generalized involution AGgroup, I(N) the involution part of N (deﬁned as Theorem 9). Then, f: N→N/I(N) is a homomorphism mapping, and N/I(N) is involutive, where N/I(N) = {[x] x∈N}, [x] is the equivalent class contained x by closed ﬁlter I(N). Proof. It follows from Theorem 10 and Theorem 11. Theorem 13. Let (N, *) be an AGgroup, P(N) = {x∈N∃a∈N, s.t x =a*a} be the power part of N (see Theorem 5). Then, P(N) is a closed ﬁlter of N. Proof. It is obvious that e = e ∗ e ∈ P(N). For any x∈N, x*x∈P(N). Moreover, assume that x∈P(N) and x*y∈P(N), then there exists a, b∈N such that x = a*a, x*y = b*b. Denote c = a−1 *b, where a−1 is the left inverse of a in N. Then, c*c = (a−1 *b)*(a−1 *b) = (a−1 *a−1 )*(b*b) = (a−1 *a−1 )*(x*y) = (a−1 *a−1 )*((a*a)*y) = (a−1 *a−1 )*((y*a)*a) = (a−1 *(y*a))*(a−1 *a)= (a−1 *(y*a))*e = (e*(y*a))*a−1 = (y*a)*a−1 = (a−1 *a)*y = e*y= y.
Thus, y∈P(N). It follows that P(N) is a ﬁlter of N. By Theorem 5, P(N) is a subalgebra of N, therefore, P(N) is a closed ﬁlter of N. Theorem 14. Let (N, *) be an AGgroup, P(N) the power part of N (deﬁned as Theorem 13). Then, f: N→N/P(N) is a homomorphism mapping, where N/P(N) = {[x] x∈N}, [x] is the equivalent class contained x by closed ﬁlter P(N). Proof. It follows from Theorems 11 and 13. 6. Conclusions In the paper, we give some examples of AGgroups, and obtain some new properties of AGgroups: an AGgroup is weak commutative (or alternative) if and only if it is an Abelian group; every AGgroup is a quasicancellative AGgroupoid. We introduce two new concepts of involution AGgroup and generalized involution AGgroup, establish a onetoone correspondence between involution AGgroups and Abelian groups, and construct a homomorphism mapping from generalized involution AGgroups to involution AGgroups. Moreover, we introduce the notion of ﬁlter in AGgroups, establish quotient algebra by every ﬁlter, and obtain some homomorphism theorems. Some results in this paper are expressed in Figure 2. In the future, we can investigate the combination of some uncertainty set theories (fuzzy set, neutrosophic set, etc.) and algebra systems (see [18–22]).
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Figure 2. Some results in this paper. Author Contributions: X.Z. and X.W. initiated the research; and X.Z. wrote ﬁnal version of the paper. Funding: This research was funded by National Natural Science Foundation of China grant number 61573240. Conﬂicts of Interest: The authors declare no conﬂict of interest.
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20. 21. 22.
Zhang, X.H.; Mao, X.Y.; Wu, Y.T.; Zhai, X.H. Neutrosophic filters in pseudoBCI algebras. Int. J. Uncertain. Quan. 2018, 8, 511–526. [CrossRef] Zhang, X.H.; Borzooei, R.A.; Jun, Y.B. Qﬁlters of quantum Balgebras and basic implication algebras. Symmetry 2018, 10, 573. [CrossRef] Zhan, J.M.; Sun, B.Z.; Alcantud, J.C.R. Covering based multigranulation (I, T)fuzzy rough set models and applications in multiattribute group decisionmaking. Inf. Sci. 2019, 476, 290–318. [CrossRef] © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Isoperimetric Numbers of Randomly Perturbed Intersection Graphs Yilun Shang Department of Computer and Information Sciences, Faculty of Engineering and Environment, Northumbria University, Newcastle NE1 8ST, UK; [email protected]; Tel.: +4401912273562 Received: 2 February 2019; Accepted: 27 March 2019; Published: 1 April 2019
Abstract: Social networks describe social interactions between people, which are often modeled by intersection graphs. In this paper, we propose an intersection graph model that is induced by adding a sparse random bipartite graph to a given bipartite graph. Under some mild conditions, we show that the vertex–isoperimetric number and the edge–isoperimetric number of the randomly perturbed intersection graph on n vertices are Ω(1/ ln n) asymptomatically almost surely. Numerical simulations for small graphs extracted from two realworld social networks, namely, the board interlocking network and the scientiﬁc collaboration network, were performed. It was revealed that the effect of increasing isoperimetric numbers (i.e., expansion properties) on randomly perturbed intersection graphs is presumably independent of the order of the network. Keywords: isoperimetric number; random graph; intersection graph; social network
1. Introduction Complex largescale network structures arise in a variety of natural and technological settings [1,2], and they pose numerous challenges to computer scientists and applied mathematicians. Many interesting ideas in this area come from the analysis of social networks [3], where each vertex (actor) is associated with a set of properties (attributes), and pairs of sets with nonempty intersections correspond to edges in the network. Complex and social networks represented by such intersection graphs are copious in the real world. Wellknown examples include the ﬁlm actor network [4], where actors are linked by an edge if they performed in the same movie, the academic coauthorship network [5], where two researchers are linked by an edge if they have a joint publication, the circle of friends in online social networks (e.g., Google+), where two users are declared adjacent if they share a common interest, and the Eschenauer–Gligor key predistribution scheme [6] in secure wireless sensor networks, where two sensors establish secure communication over a link if they have at least one common key. Remarkably, it was shown in Reference [7] that all graphs are indeed intersection graphs. To understand statistical properties of intersection graphs, a probability model was introduced in References [8,9] as a generalization of the classical model G (n, p) of Erd˝os and Rényi [10]. Formally, let n, m be positive integers and let p ∈ [0, 1]. We start with a random bipartite graph B(n, m, p) with independent vertex sets V = {v1 , · · · , vn } and W = {w1 , · · · , wm } and edges between V and W existing independently with probability p. In terms of social networks, V is interpreted as a set of actors and W a set of attributes. We then deﬁne the random intersection graph G (n, m, p) with vertex set V and vertices vi , v j ∈ V adjacent if and only if there exists some w ∈ W such that both vi and v j are adjacent to w in B(n, m, p). Several variant models of random intersection graphs have been proposed, and many graphtheoretic properties of G (n, m, p), such as degree distribution, connected components, ﬁxed subgraphs, independence number, clique number, diameter, Hamiltonicity and clustering, have been extensively studied [8,9,11–14]. We refer the reader to References [15,16] for an updated review of recent results in this proliﬁc ﬁeld.
Symmetry 2019, 11, 452; doi:10.3390/sym11040452
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In light of the above list of properties studied, it is, perhaps, surprising that there has been little work regarding isoperimetric numbers of random intersection graphs. The isoperimetric numbers, which measure the expansion properties of a graph (see Section 2 below for precise deﬁnitions), have a long history in random graph theory [17–19] and are strongly related to the graph spectrum and expanders [20]. They have found a wide range of applications in theoretical computer science, including algorithm design, data compression, rapid mixing, error correcting codes, and robust computer networks [21]. Social networks such as coauthorship networks are commonly believed to have poor expansion properties (i.e., small isoperimetric numbers), which indicate the existence of bottlenecks (e.g., cuts with small size) inside the networks, because of their modular and community organization [22,23]. In this paper, we hope to show that it is possible to increase the isoperimetric numbers by a gentle perturbation of the original bipartite graph structure underlying the intersection graphs. In recent times, there has been an effort to study the effect of random perturbation on graphs. The most mathematically famous example is perhaps the Newman–Watts smallworld network [1,24], which is a random instance obtained by adding random edges to a cycle, exhibiting short average distance and high clustering coefficient, namely, the socalled smallworld phenomenon. A random graph model G ∪ R [25] with general connected base graph G on n vertices and R being a sparse Erd˝osRényi random graph G (n, ε/n) where ε > 0 is some small constant has been introduced in [26], and its further properties, such as connectivity, ﬁxed subgraphs, Hamiltonicity, diameter, mixing time, vertex and edge expansion, have been intensively examined; see, e.g., [27–34] and references therein. For instance, in Reference [29], a necessary condition for the base graph is given under which the perturbed graph G ∪ R is an expander a.a.s. (asymptomatically almost surely); for a connected base graph G, it is shown in Reference [30] that, a.a.s. the perturbed graph has an edge–isoperimetric number Ω(1/ ln n), diameter O(ln n), and vertex–isoperimetric number Ω(1/ ln n), where for the last property G is assumed to have bounded maximum degree. Here, we say that G ∪ R possesses a graph property P asymptotically almost surely, or a.a.s. for brevity, if the probability that G ∪ R possesses P tends to 1 as n goes to inﬁnity. In this paper, to go a step further in this line of research, we investigate the bipartite graph type perturbation, where random edges are only added to the base (bipartite) graph between the two independent sets. We provide lower bounds for the isoperimetric numbers of random intersection graphs induced by such perturbations. The rest of the paper is organized as follows. In Section 2, we state and discuss the main results, with proofs relegated to Section 4. In Section 3, we give numerical examples based upon real network data, complementing our theoretical results in small network sizes. Section 5 contains some concluding remarks. 2. Results Let G = (V, E) be a graph with vertex set V and edge set E. If S ⊆ V is a set of vertices, then ∂G S denotes the set of edges of G having one end in S and the other end in V \S. Given S ⊆ V, write G [S] for the subgraph of G induced by S. We use NG (S) to denote the collection of vertices of V \S which are adjacent to some vertex of S. For a vertex v ∈ V, NG (v) is the neighborhood of v, and we denote by NG2 (v) = NG ( NG (v)) the second neighborhood of v. The above subscript G will be omitted when no ambiguity may arise. For a graph G, its edge–isoperimetric number, c( G ) (also called its Cheeger constant), is given by: ∂G S . c( G ) = min S ⊆V S 0 1. Then there exists some constant δ > 0 satisfying ι( G ( B ∪ R)) ≥ δ/ ln n a.a.s. A couple of remarks are in order. Remark 1. The local ef fects of the perturbation are quite mild, as a small ε is of interest. Nonetheless, the global inﬂuence on the vertex–isoperimetric number can be prominent. To see this, note that any connected (intersection) graph G has ι( G ) = Ω(1/n). In particular, if G is a tree, we have ι( G ) = Θ(1/n) (see e.g., [36]). Remark 2. It is easy to check that the maximum degree of G ( B) is Δ. In fact, v ∈ V and v1 ∈ V are adjacent in G ( B) if and only if they have a common neighbor w ∈ W, namely, w ∈ NB (v) and v1 ∈ NB (w). Hence, the degree of v is NB ( NB (v)). The assumption that Δ is a constant cannot be removed in general. Indeed, when α ≥ 1, consider the bipartite graph B(V, W, E) with V = {v1 , · · · , vn }, W = {w1 , · · · , wm }, and the edge set E = {{v1 , wi }, {v j , w j−1 }i = 1, · · · , n − 1, j = 2, · · · , n}. It is clear that G ( B) is a star with center v1 over the vertex set V. There are no more than n2 p edges over V \{v1 } in the graph G ( B ∪ R), which covers at most 2n2 p vertices. In G ( B ∪ R), there will be an independent set S (meaning that G ( B ∪ R)[S] is empty) of order at least: % n n − 2n2 p = n 1 − 2ε m √ and NG( B∪ R) (S) = 1. Therefore, ι( G ( B ∪ R)) ≤ 1/ n(1 − 2ε n/m) = O(1/n). When α < 1, consider the bipartite graph B(V, W, E) with the edge set E = {{v1 , wi }, {v j , w j−1 }, {vl , wm } i = 1, · · · , m, j = 2, · · · , m + 1, l = m + 2, · · · , n}. Then G ( B) can be thought of as the joining of a star K1,m having center v1 and a complete graph Kn−m+1 by identifying v1 with any vertex of Kn−m+1 . After adding nmε/n = mε edges to B, in G ( B ∪ R), there will be an independent set S of order at least m − 1 − 2mε and NG( B∪ R) (S) = 1. Therefore, ι( G ( B ∪ R)) ≤ 1/(m − 1 − 2mε) = O(1/m). Recall that the inequality c( G ) ≥ ι( G ) holds for any graph G. Therefore, a direct corollary of Theorem 1 reads c( G ( B ∪ R)) ≥ δ/ ln n a.a.s. for some δ > 0. The following theorem shows that this lower bound for edge–isoperimetric number actually holds without any assumption on Δ.
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Theorem 2. Let B = B(V, W, E) be a bipartite graph with V  = n and W  = m = nα such that any two √ vertices in V are connected by a path. For any ε > 0, let R ∼ B(n, m, p) with p = ε/ nm. Then there exists some constant δ > 0 satisfying c( G ( B ∪ R)) ≥ δ/(1 + ln n) a.a.s. Theorems 1 and 2 hold in the sense of large n limit. In the next section, we shall demonstrate that the isoperimetric numbers can be improved as well for small randomly perturbed intersection graphs based upon real network data. 3. Illustration on Small Networks To ﬁnd the exact isoperimetric numbers, one needs to calculate the minimum fraction of neighboring vertices or edges over the nodes inside the subset for all possible subsets of vertices with order at most V /2. Since this is an NPhard problem, it is intractable to compute the exact values for general graphs [21,35]. It is well known that Cheeger’s inequality, also known as the Alon–Milman inequality, provides bounds for the isoperimetric numbers using graph Laplacian eigenvalues. On the other hand, standard algorithms in linear algebra can be used to efficiently compute the spectrum of a given large graph. Here, instead of evaluating “approximate” values involving other parameters such as eigenvalues, we are interested in obtaining exact values of ι( G ( B ∪ R)) and c( G ( B ∪ R)) for small networks. Two intersectionbased social networks are considered here: (i) The Norwegian interlocking directorate network NorBoards [37], where two directors are adjacent if they are sitting on the board of the same company based on the Norwegian Business Register on 5 August 2009. The underlying ¯ E¯ ) contains V¯  = 1495 directors, W ¯  = 367 companies, and  E¯  = 1834 ¯ W, bipartite graph B¯ (V, edges indicating the affiliation relations; (ii) the coauthorship network caCondMat [5] based on preprints posted to the Condensed Matter Section of arXiv EPrint Archive between 1995 and 1999. ¯ E¯ ) contains V¯  =16,726 authors, W ¯  = 22,016 papers, ¯ W, The underlying bipartite graph B¯ (V, and  E¯  = 58,596 edges indicating authorship. Figures 1 and 2 report the vertex–isoperimetric numbers and edge–isoperimetric numbers for subsets of NorBoards and caCondMat, respectively. For a given n ∈ [20, 30], we ﬁrst take ¯ E¯ ) with V  = n so that G ( B) is connected and calculate its ¯ W, a subgraph B = B(V, W, E) from B¯ (V, vertex–isoperimetric and edge–isoperimetric numbers. Each data point (blue square) in Figures 1 and 2 is obtained by means of an ensemble averaging of 30 independently taken graphs. For each chosen bipartite graph B, we then perturb it following the rules speciﬁed in Theorems 1 and 2 with ε = 1 to get the perturbed intersection graph G ( B ∪ R). Each data point (red circle) in Figures 1 and 2 is obtained by means of a mixed ensemble averaging of 50 independentlyimplemented perturbations for 30 graphs. From a statistics viewpoint, it is clear that our random perturbation scheme increases both the vertex–isoperimetric and the edge–isoperimetric number for both cases. This, together with the theoretical results, suggests that the quantitative effect of random perturbations is independent of the order of the network. Remark 3. It is worth stressing that the theoretical results (Theorems 1 and 2) are in the large limit of the network size n. In other words, the form ln1n only makes sense as n → ∞. The simulation results presented in Figures 1 and 2 are for very small networks. Therefore, these results have no bearing on the ln1n dependence (although a slight decline tendency for ι( G ( B ∪ R)) can be seen in Figure 1a). The main phenomenon we observe from Figures 1 and 2 is that the random perturbation increases both vertex– and edge–isoperimetric numbers for all the cases considered. The numerical results (for small ﬁnite graphs) are a nice complement to the theoretical results (for inﬁnite graphs). However, our numerical observations neither prove the ln1n dependence would hold for small graphs nor show that such an increase of isoperimetric numbers would be universal in any sense. (A practical issue stems from graph sampling. To establish a proper model ﬁt to the data, Akaike information criteria and Bayesian information criteria need to be applied.) The establishment of correlation between isoperimetric numbers and graph size n for ﬁnite intersection graphs is an interesting future work.
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(a)
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Figure 1. Vertex–isoperimetric number (panel (a)) and edge–isoperimetric number (panel (b)) versus n =  G ( B) for subgraphs G ( B) (and its randomly perturbed version G ( B ∪ R)) taken from NorBoards. (a)
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n
Figure 2. Vertex–isoperimetric number (panel (a)) and edge–isoperimetric number (panel (b)) versus n =  G ( B) for subgraphs G ( B) (and its randomly perturbed version G ( B ∪ R)) taken from caCondMat.
4. Proofs In this section, we prove Theorems 1 and 2. Our idea behind this is somewhat simple: If the network can be carefully decomposed into some subnetworks so that the resulting supernetwork (with these subnetworks being supervertices) is sparse and highly connected, then its isoperimetric numbers are expected to be high. Similar approaches have been applied in, e.g., References [29–31]. Proof of Theorem 1. Set s = CΔ(ln n)/ε for some constant C = C (ε) > 0 to be determined. By assumption, G ( B) is connected. Following Reference [38] (Proposition 4.5), we can divide the vertex set V into disjoint sets V1 , V2 , · · · , Vθ satisfying s ≤ Vi  ≤ Δs and G ( B)[Vi ] connected for each i. Clearly, n/(Δs) ≤ θ ≤ n/s. Let [θ ] = {1, 2, · · · , θ }. For a graph G = (V, E), we say two sets S1 , S2 ⊆ V have common neighbors in G if there exist v1 ∈ S1 , v2 ∈ S2 , and v ∈ V such that {v1 , v} ∈ E and {v2 , v} ∈ E hold. We will ﬁrst show the following property for the random bipartite graph R holds a.a.s.: For every Θ ⊆ [θ ] with 0 < Θ ≤ θ/2, there exist at least Θ/2 many of Vi (i ∈ [θ ]\Θ) which have common neighbors with ∪i∈Θ Vi in R. Indeed, the probability that two sets Vi and Vj have no common neighbors in R can be computed # $m as 1 − [1 − (1 − p)Vi  ][1 − (1 − p)Vj  ] . Hence, the probability that there exists a set Θ ⊆ [θ ] with 0 < Θ ≤ θ/2 such that no more than Θ/2 many of Vi (i ∈ [θ ]\Θ) have common neighbors with ∪i∈Θ Vi in R is upper bounded by:
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Symmetry 2019, 11, 452
≤ where (θj )(
θ−j j ) 2
$m θ−j # θ ∑ j j 1 − [1 − (1 − p) js ][1 − (1 − p)(θ−3j/2)s ] , 1≤ j≤θ/2 2 m j s2 j(θ − 3j/2) p2 eθ 2e(θ − j) j/2 1 − , ∑ j j 4 1≤ j≤θ/2
counts the choice of Θ (with Θ = j) and the corresponding sets {Vi } described
above, the estimate Vi  ≥ s for all i ∈ [θ ] is utilized in the multiplicative probabilities (i.e., there are at least (θ − 3j/2) sets in the union ∪i∈Θ Vi ), and the upper bound comes from a direct application of inequalities ([10], p. 386). The above probability is further upper bounded by (C (ln n)/n)m ∑1≤ j≤θ/2 (2θ/j)3j/2 = o (1) when α ≤ 1, and is upper bounded by ∑1≤ j≤θ/2 θ 3j/2+2 exp(−CεΔj ln n) = o (1) when α > 1 for a sufficiently large C. Therefore, the above property for the random bipartite graph R holds a.a.s. In the following, we will condition on such an R. Fix a set S ⊆ V with S ≤ n/2. Deﬁne three sets of indices: Θ0 = {i ∈ [θ ]Vi ⊆ S}, Θ1 = {i ∈ [θ ]0 < Vi ∩ S < Vi }, and Θ2 = {i ∈ [θ ]\Θ0  NG( B∪ R) (Vi ) ∩ S = ∅}. Note that Θ0 and Θ1 are deterministic, but Θ2 is a random set. If Θ0  ≤ θ/2, Θ2  ≥ Θ0 /2 a.a.s. by the above assumed property of R. Similarly, if Θ0  > θ/2, we have Θ2  ≥ Θ/2 = (θ − Θ0  − Θ2 )/2 a.a.s., where Θ = [θ ]\(Θ0 ∪ Θ2 ). Hence, Θ2  ≥ min{Θ0 /2, (θ − Θ0 )/3} a.a.s. Recall that S ≤ n/2. We derive that n/2 ≤ V \S ≤  ∪i ∈Θ0 Vi  ≤ (θ − Θ0 )Δs ≤ (θ − Θ0 )Δn/θ, and thus, θ − Θ0  ≥ θ/(2Δ). Therefore, we have a.a.s.:  Θ0  θ Θ  Θ2  ≥ min ≥ 0 . , 2 6Δ 6Δ By deﬁnition, we have S ⊆ ∪i∈Θ0 ∪Θ1 Vi . Thus, S ≤ (Θ0  + Θ1 )Δs. Since G ( B)[Vi ] for i ∈ Θ1 is connected,  NG( B∪ R) (S) ≥ Θ1 ∪ Θ2 . Now we consider two cases. If Θ1  ≥ Θ0 , then  NG( B∪ R) (S) ≥ Θ1  ≥ S/(2Δs). If Θ1  ≤ Θ0 , then  NG( B∪ R) (S) ≥ Θ2  ≥ Θ0 /(6Δ) ≥ S/(12Δ2 s) a.a.s. Therefore:
 NG( B∪ R) (S) ≥ min S
1 1 , 2Δs 12Δ2 s
a.a.s.
Recall the deﬁnition of s at the beginning of the proof, and we complete the proof by taking δ = ε/(12Δ3 C ). We have made no attempt to optimize the constants in the proof. It is easy to check that the condition that G ( B) is connected in Theorem 1 can be weakened. For example, the above proof holds if each connected component of G ( B) is of order at least CΔ(ln n)/ε. Let G = (V, E) be a graph of order n. For integers a, b, and c, deﬁne S( a, b, c) as a collection of all sets S ⊆ V such that S = a and there exists a partition S = S1 ∪ · · · ∪ Sb , where each G [Si ] is connected, there are no edges in E connecting different Si , and  NG (S1 ) + · · · +  NG (Sb ) = c. The next lemma gives an upper bound of the size of S( a, b, c). Lemma 1. ([30])
S( a, b, c) ≤
en b ea b ec b e( a + c) c b
b
b
c
.
Proof of Theorem 2. Consider the family S( a, b, c) of sets deﬁned in graph G ( B). Since G ( B) is connected, we have for each S ∈ S( a, b, c), ∂G( B) S ≥ c ≥ b. Note that ∂G( B∪ R) S ≥ ∂G( R) S
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holds. It suffices to show that the following property for the random bipartite graph R holds a.a.s.: There are constants K, δ > 0 such that for any K ln n ≤ a ≤ n/2, we have:
 ∂ G ( R) S  ≥
δa , 1 + ln n
for each S ∈ S( a, b, c) with b ≤ c ≤ δa/(1 + ln n). Indeed, when S = a ≤ K ln n, we can choose a small δ such that 2Kδ ≤ 1. Thus, ∂G( B∪ R) S ≥ ∂G( B) (S) ≥ 1 ≥ δa/(1 + ln n). It follows from Lemma 1 and b ≤ c ≤ δa/(1 + ln n) ≤ a that:
S( a, b, c) ≤
2e4 na2 c3
c
≤
2e4 n(1 + ln n)3 δ3 a
δa/(1+ln n)
≤ eCδa ln(1/δ) ,
for some constant C > 0, where the ﬁrst inequality holds since f ( x ) = (eρ/x ) x is increasing on (0, ρ] and the second inequality holds since g( x ) = (ρ/x3 ) x is increasing on (0, ρ1/3 ]. Note that mp2 → 0 and 1 − (1 − p2 )m ∼ mp2 . For a ﬁxed S with S = a ≤ n/2, we obtain: na P(∂G( R) S < δa) P(Bin( a(n − a), mp2 ) < δa) ≤ P Bin , mp2 < δa 2 aε2 ≤ exp − , 16 provided δ < ε2 /4, where the ﬁrst inequality relies on Reference [9] (Theorem 2.2) and the last line uses a standard Chernoff’s bound (e.g., [10]). Hence: δa δa n P  ∂ G ( R) S  < , ∃S ∈ S( a, b, c), b ≤ c ≤ , K ln n ≤ a ≤ 1 + ln n 1 + ln n 2 ≤ P ∂G( R) S < δa, ∃S ∈ S( a, b, c), b ≤ c ≤ n, K ln n ≤ a ≤ n 1 aε2 ≤ n3 exp Cδa ln − . δ 16 By taking Cδ ln(1/δ) ≤ ε2 /32 and K ≥ 100/ε2 , the last line above is upper bounded by n3 exp(−ε2 a/32) ≤ n3 exp(−ε2 K (ln n)/32) ≤ n3 exp(−25(ln n)/8) = o (1) as n → ∞. The proof is complete. 5. Concluding Remarks In this paper, we presented a model of randomly perturbed intersection graphs. The intersection graph is induced by a given bipartite graph (base graph) plus a binomial random bipartite graph. We proved that a.a.s., the vertex–isoperimetric number and the edge–isoperimetric number of the randomly perturbed intersection graphs are of order Ω(1/ ln n) under some mild conditions. It would be interesting to investigate path length, diameter, and clustering coefficient of this model, which are important characteristics of reallife complex and social networks. Another intriguing direction is to examine more general intersection graph models, such as active and passive intersection graphs [39]. In particular, if two vertices in one independent set V are declared adjacent when they have at least k ≥ 1 common neighbors in the other independent set W, what role will k play in estimating the isoperimetric numbers, clustering, and path length of the resulting perturbed intersection graphs? Other perturbation mechanisms are also of research interest. Acknowledgments: I am very grateful to the editor and two anonymous referees for their valuable comments that have greatly improved the presentation of the paper. The author was supported by a Starting Grant of Northumbria University. Conﬂicts of Interest: The author declares no conﬂict of interest.
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Generalized Permanental Polynomials of Graphs Shunyi Liu School of Science, Chang’an University, Xi’an 710064, China; [email protected] Received: 28 January 2019; Accepted: 14 February 2019; Published: 16 February 2019
Abstract: The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to ﬁnd a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A( G ) and degree matrix D ( G ). The generalized permanental polynomial of G is deﬁned by PG ( x, μ) = per( xI − ( A( G ) − μD ( G ))). In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efﬁcient for distinguishing graphs. Furthermore, we can write PG ( x, μ) in the coefﬁcient form ∑in=0 cμi ( G ) x n−i and obtain the combinatorial expressions for the ﬁrst ﬁve coefﬁcients cμi ( G ) (i = 0, 1, . . . , 4) of PG ( x, μ). Keywords: generalized permanental polynomial; coefﬁcient; copermanental
1. Introduction A graph invariant f is a function from the set of all graphs into any commutative ring, such that f has the same value for any two isomorphic graphs. Graph invariants can be used to check whether two graphs are not isomorphic. If a graph invariant f satisﬁes the condition that f ( G ) = f ( H ) implies G and H are isomorphic, then f is called a complete graph invariant. The problem of ﬁnding complete graph invariants is closely related to the graph isomorphism problem. Up to now, no complete graph invariant for general graphs has been found. However, some complete graph invariants have been identiﬁed for special cases and graph classes (see, for example, [1]). Graph polynomials are graph invariants whose values are polynomials, which have been developed for measuring the structural information of networks and for characterizing graphs [2]. Noy [3] surveyed results for determining graphs that can be characterized by graph polynomials. In a series of papers [1,4–6], Dehmer et al. studied highly discriminating descriptors to distinguish graphs (networks) based on graph polynomials. In [5], it was found that the graph invariants based on the zeros of permanental polynomials are quite efﬁcient in distinguishing graphs. Balasubramanian and Parthasarathy [7,8] introduced the bivariate permanent polynomial of a graph and conjectured that this graph polynomial is a complete graph invariant. In [9], Liu gave counterexamples to the conjecture by a computer search. In order to ﬁnd almost complete graph invariants, we introduce a graph polynomial by employing graph matrices and the permanent of a square matrix. We will see that this graph polynomial turns out to be quite efﬁcient when we use it to distinguish graphs (networks). The permanent of an n × n matrix M with entries mij (i, j = 1, 2, . . . , n) is deﬁned by per( M ) =
Symmetry 2019, 11, 242; doi:10.3390/sym11020242
n
∑ ∏ miσ(i) , σ i =1
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Symmetry 2019, 11, 242
where the sum is over all permutations σ of {1, 2, . . . , n}. Valiant [10] proved that computing the permanent is #Pcomplete, even when restricted to (0,1)matrices. The permanental polynomial of M, denoted by π ( M, x ), is deﬁned to be the permanent of the characteristic matrix of M; that is, π ( M, x ) = per( xIn − M), where In is the identity matrix of size n. Let G = (V ( G ), E( G )) be a graph with adjacency matrix A( G ) and degree matrix D ( G ). The Laplacian matrix and signless Laplacian matrix of G are deﬁned by L( G ) = D ( G ) − A( G ) and Q( G ) = D ( G ) + A( G ), respectively. The ordinary permanental polynomial of a graph G is deﬁned as the permanental polynomial of the adjacency matrix A( G ) of G (i.e., π ( A( G ), x )). We call π ( L( G ), x ) (respectively, π ( Q( G ), x )) the Laplacian (respectively, the signless Laplacian) permanental polynomial of G. The permanental polynomial π ( A( G ), x ) of a graph G was ﬁrst studied in mathematics by Merris et al. [11], and it was ﬁrst studied in the chemical literature by Kasum et al. [12]. It was found that the coefﬁcients and roots of π ( A( G ), x ) encode the structural information of a (chemical) graph G (see, e.g., [13,14]). Characterization of graphs by the permanental polynomial has been investigated, see [15–19]. The Laplacian permanental polynomial of a graph was ﬁrst considered by Merris et al. [11], and the signless Laplacian permanental polynomial was ﬁrst studied by Faria [20]. For more on permanental polynomials of graphs, we refer the reader to the survey [21]. We consider a bivariate graph polynomial of a graph G on n vertices, deﬁned by PG ( x, μ) = per( xIn − ( A( G ) − μD ( G ))). It is easy to see that PG ( x, μ) generalizes some wellknown permanental polynomials of a graph G. For example, the ordinary permanental polynomial of G is PG ( x, 0), the Laplacian permanental polynomial of G is (−1)V (G) PG (− x, 1), and the signless Laplacian permanental polynomial of G is PG ( x, −1). We call PG ( x, μ) the generalized permanental polynomial of G. We can write the generalized permanental polynomial PG ( x, μ) in the coefﬁcient form PG ( x, μ) =
n
∑ cμi (G)xn−i .
i =0
The general problem is to achieve a better understanding of the coefﬁcients of PG ( x, μ). For any graph polynomial, it is interesting to determine its ability to characterize or distinguish graphs. A natural question is how well the generalized permanental polynomial distinguishes graphs. The rest of the paper is organized as follows. In Section 2, we obtain the combinatorial expressions for the ﬁrst ﬁve coefﬁcients cμ0 , cμ1 , cμ2 , cμ3 , and cμ4 of PG ( x, μ), and we compute the ﬁrst ﬁve coefﬁcients of PG ( x, μ) for some speciﬁc graphs. In Section 3, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The presented data shows that the generalized permanental polynomial is quite efﬁcient in distinguishing graphs. It may serve as a powerful tool for dealing with graph isomorphisms. 2. Coefﬁcients In Section 2.1, we obtain a general relation between the generalized and the ordinary permanental polynomials of graphs. Explicit expressions for the ﬁrst ﬁve coefﬁcients of the generalized permanental polynomial are given in Section 2.2. As an application, we obtain the explicit expressions for the ﬁrst ﬁve coefﬁcients of the generalized permanental polynomials of some speciﬁc graphs in Section 2.3.
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2.1. Relation between the Generalized and the Ordinary Permanental Polynomials First, we present two properties of the permanent. Lemma 1. Let A, B, and C be three n × n matrices. If A, B, and C differ only in the rth row (or column), and the rth row (or column) of C is the sum of the rth rows (or columns) of A and B, then per(C ) = per( A) + per( B). Lemma 2. Let M = (mij ) be an n × n matrix. Then, for any i ∈ {1, 2, . . . , n}, per( M) =
n
∑ mij per( M(i, j)),
j =1
where M(i, j) denotes the matrix obtained by deleting the ith row and jth column from M. Since Lemmas 1 and 2 can be easily veriﬁed using the deﬁnition of the permanent, the proofs are omitted. We need the following notations. Let G = (V ( G ), E( G )) be a graph with vertex set V ( G ) = {v1 , v2 , . . . , vn } and edge set E( G ). Let di = dG (vi ) be the degree of vi in G. The degree matrix D ( G ) of G is the diagonal matrix whose (i, i )th entry is dG (vi ). Let vr1 , vr2 , . . . , vrk be k distinct vertices of G. Then Gr1 ,r2 ,...,rk denotes the subgraph obtained by deleting vertices vr1 , vr2 , . . . , vrk from G. We use G [ hr ] to denote the graph obtained from G by attaching to the vertex vr a loop of weight hr . Similarly, G [ hr , hs ] stands for the graph obtained by attaching to both vr and vs loops of weight hr and hs , respectively. Finally, G [ h1 , h2 , . . . , hn ] is the graph obtained by attaching a loop of weight hr to vertex vr for each r = 1, 2, . . . , n. The adjacency matrix A( G [ hr1 , hr2 , . . . , hrs ]) of G [ hr1 , hr2 , . . . , hrs ] is deﬁned as the n × n matrix ( aij ) with ⎧ ⎪ ⎨ hr , if i = j = r and r ∈ {r1 , r2 , . . . , rs }, aij = 1, if i = j and vi v j ∈ E( G ), ⎪ ⎩ 0, otherwise. By Lemmas 1 and 2, expanding along the rth column, we can obtain the recursion relation π ( A( G [ hr ]), x ) = π ( A( G ), x ) − hr π ( A( Gr ), x ).
(1)
For example, expanding along the ﬁrst column of π ( A( G [ h1 ]), x ), we have π ( A( G [ h1 ]), x ) = per( xIn − A( G [ h1 ])) , x − h1 u = per v xIn−1 − A( G1 ) , , x u − h1 = per + per v xIn−1 − A( G1 ) 0
u xIn−1 − A( G1 )
= π ( A( G ), x ) − h1 per( xIn−1 − A( G1 )) = π ( A( G ), x ) − h1 π ( A( G1 ), x ). By repeated application of (1) for G [ hr , hs ], we have π ( A( G [ hr , hs ]), x )
= π ( A( G [hr ]), x ) − hs π ( A( Gs [hr ]), x ) = π ( A( G ), x ) − hr π ( A( Gr ), x ) − hs (π ( A( Gs ), x ) − hr π ( A( Gr,s ), x )) = π ( A( G ), x ) − hr π ( A( Gr ), x ) − hs π ( A( Gs ), x ) + hr hs π ( A( Gr,s ), x ).
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Additional iterations can be made to take into account loops on additional vertices. For loops on all n vertices, the expression becomes π ( A( G [ h1 , h2 , . . . , hn ]), x ) = π ( A( G ), x ) +
n
∑ (−1)k
k =1
∑
1≤r1 0 is the regularization parameter, and uα,δ is the regularized Laplacian operation. Assuming that hα,δ is a function satisfying ( Δhα,δ = uα,δ , in Ω , on ∂Ω hα,δ = 0, then it has A[uα,δ ] = − hα,δ . Equation (3) can be rewritten as (
αΔhα,δ − hα,δ = − f δ , in Ω hα,δ = 0, on ∂Ω
(4)
This boundary value problem of PDE can be solved by classic numerical methods, and then the regularized Laplacian operation uα,δ can be expressed as uα,δ (r ) = Δhα,δ (r ) =
1 α,δ [h (r ) − f δ (r )], r ∈ Ω. α
(5)
From the above rewriting, we can see that (4) and (5) are equivalent to the integral Equation (3). Compared with solving the regularization Equation (3) directly, the computational burden of solving (4) and (5) is reduced drastically. The work of [16] mainly focuses on the choice of the regularization parameter α and the error estimate of the regularized Laplacian operation uα,δ . Unfortunately, the choice strategy given in [16] depends on the noise level of the noise data, which is unknown in practice. Since the choice strategy of the regularization parameter plays an important role in the regularization method, as the authors stated in [16], the selection of parameter α in the edge detection algorithm should be considered carefully. 3. The Edge Detection Algorithm In this section, we will construct the novel edge detection algorithm based on the regularized Laplacian operation given in Section 2. The ﬁrst thing we are concerned with is the weakness of the Lavrentiev regularization. Notice that hα,δ (r ) = 0, r ∈ ∂Ω, it has uα,δ (r ) = − α1 f δ (r ), r ∈ ∂Ω. The parameter α > 0 is usually a small number, which means the error of the regularized Laplacian operation on the boundary can be ampliﬁed α1 times. Thus, the computation is meaningless on ∂Ω. In fact, the validity of the regularized Laplacian operation uα,δ (r ) has been weakened when r is close to the boundary. Experiments in [16] have shown
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that the weakness only affects the points very close to the boundary. Hence, except a few pixels which are as close as possible to the boundary of the image domain, the edge detection results will be acceptable. The second thing we are concerned with is the choice strategy of the regularization parameter α. Since the noise level of an image data is unknown, the choice strategy given in [16] cannot be carried out. Considering only the edge detection problem, the objective parameter selection given in [17] can be adopted to choose the regularization parameter. Once the regularization parameter α is chosen, the regularized Laplacian operation uα,δ can be obtained by solving Equations (4) and (5), where Equation (4) can be solved by the standard ﬁnite difference method or ﬁnite element method. Combined with the objective parameter selection given in [17], the main framework of the choice strategy is summarized as follows: Step 1: Regularization parameters α j , j ∈ {1, 2, . . . , n} are used to generate N different edge maps D j , j ∈ {1, 2, . . . , n} by the proposed edge detection algorithm. Then, N potential ground truths (PGTs) are constructed, and each PGTi includes pixels which have been identiﬁed as edges by at least i different edge maps. Step 2: Each PGTi is compared with each edge map Dj , and it generates four different probabilities:TPPGTi , Dj , FPPGTi , Dj , TNPGTi , Dj , FNPGTi , Dj . Among them, TPA,B (true positive) means the probability of pixels which have been determined as edges in both edge maps A and B; FPA,B (false positive) means the probability of pixels determined as edges in A, but nonedges in B; TNA,B (true negative) means the probability of pixels determined as nonedges in both A and B; and FNA,B (false negative) means the probability of pixels determined as edges in B, but nonedges in A. Step 3: For each PGTi , we average the four probabilities over all edge maps Dj , and get TPPGTi , FPPGTi , TNPGTi , FNPGTi , where TPPGTi =
1 N
N
∑ TPPGTi ,Dj , and the expressions of other
j =1
probabilities are similar. Then, a statistical measurement of each PGTi is given by the Chisquare test: χ2PGTi =
TPR − Q (1 − FPR) − (1 − Q) · , 1−Q Q
where Q = TPPGTi + FPPGTi , TPR =
TPPGTi TPPGTi + FNPGTi
, FPR =
(6)
FPPGTi FPPGTi + TNPGTi
.
The PGTi with the highest χ2PGTi is considered as the estimated ground truth (EGT). Step 4: Each edge map’s Dj is then matched to the EGT by four new probabilities: TPDj ,EGT , FPDj ,EGT , TNDj ,EGT , FNDj ,EGT . The Chisquare measurements χ2Dj are obtained by the same way as in Step 3. Then, the best edge map is the one which gives the highest χ2Dj , and the corresponding regularization parameter α j is the one we want. The Chisquare measure (6) can reﬂect the similarity of two edge maps, and the bigger the value of the Chisquare measurement, the better. As LopezMolina et al. stated in [19], the Chisquare measurement can evaluate the errors caused by spurious responses (false positives, FPs) and missing edges (false negatives, FNs), but it cannot work on the localization error when the detected edges deviate from their true position. For example, a reference edge image and three polluted edge maps are given in Figure 1. Compared with the reference edge (Figure 1a), the Chisquare measurements of the three polluted edge maps are the same, yet their localization accuracies are different. In order to reﬂect the localization error in these edge maps, distancebased error measures should be introduced.
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(a)
(b)
(c)
(d)
Figure 1. The reference edge image and three polluted edge maps: (a) reference edge ER ; (b) polluted edge map E1 ; (c) polluted edge map E2 ; (d) polluted edge map E3 .
The Baddeley’s delta metric (BDM) is one of the most common distancebased measures [20]. It has been proven to be an ideal measure for the comparison of edge detection algorithms [19,21]. Let A and B be two edge maps with the same resolution M × N, and P = {1, . . . , M} × {1, . . . , N } be the set of pixels in the image. The value of BDM between A and B is deﬁned as , Δ ( A, B) = k
1 w(d( p, A)) − w(d( p, B))k MN p∑ ∈P
1/k ,
(7)
where d( p, A) is the Euclidean distance from p ∈ P to the closest edge points in A, the parameter k is a given positive integer and w(d( p, A)) = min(d( p, A), c) for a given constant c > 0. Compared with the reference edge ER in Figure 1, the BDMs of the three polluted edge maps Ei (i = 1, 2, 3) are given in Table 1 with different parameters c and k. The smaller the value of BDM, the better. As we can see from Table 1, localization errors of the three edge maps are apparently distinguished. Therefore, the Chisquare measure (6) will be replaced by the BDM (7) in the choice strategies of the regularization parameter. Table 1. The Baddeley’s delta metrics (BDMs) between the reference edge image ER and the polluted edge maps Ei (i = 1, 2, 3) with the different choices of parameters c and k. Parameter Sets
Δk (ER ,E1 )
Δk (ER ,E2 )
Δk (ER ,E3 )
k = 1, c = 2 k = 1, c = 3 k = 1, c = 4 k = 2, c = 2 k = 2, c = 3 k = 2, c = 4
0.0566 0.0950 0.1397 0.2182 0.2753 0.3317
0.0937 0.1879 0.2614 0.3307 0.4925 0.6159
0.1256 0.2461 0.3305 0.3637 0.6313 0.8021
4. Experiments and Results In order to show the validity of the proposed edge detection algorithm, some comparative experiments are given in this section. In the experiments, our regularized edge detector (RED) will be compared with the LoG detector and the Laplacianbased edge detector (LED) proposed in [7]. As Yitzhaky and Peli said in [17], the parameter selection for edge detection depends mainly on the set of parameters used to generate the initial detection results. In order to reduce this inﬂuence properly, the range of the parameter is set to be large enough that instead of forming a very sparse edge map it forms a very dense one. The scale parameter of the LoG detector is set from 1.5 to 4 in steps of 0.25. The regularization parameter of the regularized edge detector is set from 0.01 (≈0) to 0.1 in steps of 0.01. The images we used are taken from [22], and some of them are shown in Figure 2. The optimal edge maps given in [22] will be seen as the ground truth in our quantitative comparisons.
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Let us ﬁrst consider the choice strategy of the regularization parameter α, where the parameters in BDM are set as k = 1, c = 2. Taking the airplane image as an example, the BDM of each PGTi , i ∈ {1, 2, . . . 11} is shown in Figure 3a, from which we can see the EGT is PGT6 . Compared with the EGT, the BDM of each edge map Dj is shown in Figure 3b, from which we can see the best edge map is D6 . Hence, the regularization parameter is chosen as α = 0.05. The choice of the scale parameter in the LoG detector is carried out similarly. It does not need any parameters in the LED. For the airplane image, the ground truth and edges detected by the three edge detectors are shown in Figure 4. From Figure 4b, we can see that the inﬂuence of the Lavrentiev regularization’s weakness on the RED is negligible. From Figure 4b,c, we can see that the RED is better than the LoG detector for noise suppression and maintaining continuous edges. Comparing Figure 4d with Figure 4b,c, we can see the superiority of the parameterdependent edge detector. Similar results for the elephant image are shown in Figure 5. For some images taken from [22], quantitative comparisons of the edges detected by the LoG detector, the RED and the LED against the ground truth are given in Table 2. Since the smaller the value of BDM, the better, this shows that the RED has better performance than the LoG detector and the LED in most cases.
(a)
(b)
Figure 2. Some images taken from [22]: (a) airplane; (b) elephant.
0.2
0.15
j
0.1
Δ1D
Δ1PGT
i
0.15 0.1 0.05
0.05
X: 6 Y: 0.02707
2
4
6 i
8
0
10
(a)
X: 6 Y: 0.001033
2
4
6 j
8
10
(b)
Figure 3. The ﬁgure of BDMs: (a) the BDM of Δ1PGTi , i ∈ {1, 2, . . . , 11}; (b) the BDM of Δ1Dj , i ∈ {1, 2, . . . , 11}.
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(a)
(b)
(c)
(d)
Figure 4. Edge detection results of the airplane image: (a) the ground truth; (b) the edge detected by the regularized edge detector (RED); (c) the Laplacian of Gaussian (LoG); (d) the Laplacianbased edge detector (LED).
(a)
(b)
(c)
(d)
Figure 5. Edge detection results of the elephant image: (a) the ground truth; (b) the edge detected by the RED; (c) the LoG; (d) the LED.
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Table 2. Quantitative comparison of the edges detected by the LoG, the RED and the LED. Images
LED
LoG
RED
Airplane Elephant Turtle Brush Tiger Grater Pitcher
0.7515 0.6619 0.4430 0.5790 0.9239 0.5537 0.5032
0.1270 0.3041 0.1226 0.1883 0.2854 0.2353 0.2584
0.1232 0.2593 0.1323 0.1673 0.2748 0.2143 0.2296
5. Conclusions In this paper, a novel edge detection algorithm is proposed based on the regularized Laplacian operation. The PDEbased regularization enables us to compute the regularized Laplacian operation in a direct way. Considering the importance of the regularization parameter, an objective choice strategy of the regularization parameter is proposed. Numerical implementations of the regularization parameter and the edge detection algorithm are also given. Based on the image database and ground truth edges taken from [22], the superiority of the RED against the LED and the LoG detector has been shown by the edge images and quantitative comparison. Author Contributions: All the authors inferred the main conclusions and approved the current version of this manuscript. Funding: This research was funded by National Natural Science Foundation of China (Grant No. 11661008), Natural Science Foundation of Jianxi Province (Grant No. 20161BAB211025), Science & Technology Project of Jiangxi Educational Committee (Grant No. GJJ150982) and Tendering Subject of Gannan Normal University (Grant No. 15zb03). Conﬂicts of Interest: The authors declare no conﬂict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Basu, M. Gaussianbased edgedetection methods—A survey. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 2002, 32, 252–260. [CrossRef] Marr, D.; Hildreth, E. Theory of edge detection. Proc. R. Soc. Lond. B 1980, 207, 187–217. [CrossRef] [PubMed] Canny, J. A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 1986, 8, 679–698. [CrossRef] [PubMed] Sarkar, S.; Boyer, K. Optimal inﬁnite impulse response zero crossing based edge detectors. CVGIP Image Underst. 1991, 54, 224–243. [CrossRef] Demigny, D. On optimal linear ﬁltering for edge detection. IEEE Trans. Image Process. 2002, 11, 728–737. [CrossRef] [PubMed] Kang, C.C.; Wang, W.J. A novel edge detection method based on the maximizing objective function. Pattern Recognit. 2007, 40, 609–618. [CrossRef] Wang, X. Laplacian operatorbased edge detectors. IEEE Trans. Pattern Anal. Mach. Intell. 2007, 29, 886–890. [CrossRef] [PubMed] LopezMolina, C.; Bustince, H.; Fernandez, J.; Couto, P.; De Baets, B. A gravitational approach to edge detection based on triangular norms. Pattern Recognit. 2010, 43, 3730–3741. [CrossRef] Murio, D.A. The Molliﬁcation Method and the Numerical Solution of IllPosed Problems; WileyInterscience: New York, NY, USA, 1993; pp. 1–5, ISBN 0471594083. Wan, X.Q.; Wang, Y.B.; Yamamoto, M. Detection of irregular points by regularization in numerical differentiation and application to edge detection. Inverse Probl. 2006, 22, 1089–1103. [CrossRef] Xu, H.L.; Liu, J.J. Stable numerical differentiation for the second order derivatives. Adv. Comput. Math. 2010, 33, 431–447. [CrossRef] Huang, X.; Wu, C.; Zhou, J. Numerical differentiation by integration. Math. Comput. 2013, 83, 789–807. [CrossRef]
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13. 14. 15. 16. 17. 18. 19. 20. 21.
22.
Wang, Y.C.; Liu, J.J. On the edge detection of an image by numerical differentiations for gray function. Math. Methods Appl. Sci. 2018, 41, 2466–2479. [CrossRef] Gonzalez, R.C.; Woods, R.E. Digital Image Processing, 3rd ed.; Pearson: London, UK, 2007; pp. 158–162, ISBN 9780131687288. Gunn, S.R. On the discrete representation of the Laplacian of Gaussian. Pattern Recognit. 1999, 32, 1463–1472. [CrossRef] Xu, H.L.; Liu, J.J. On the Laplacian operation with applications in magnetic resonance electrical impedance imaging. Inverse Probl. Sci. Eng. 2013, 21, 251–268. [CrossRef] Yitzhaky, Y.; Peli, E. A method for objective edge detection evaluation and detector parameter selection. IEEE Trans. Pattern Anal. Mach. Intell. 2003, 25, 1027–1033. [CrossRef] Gu, C.H.; Li, D.Q.; Chen, S.X.; Zheng, S.M.; Tan, Y.J. Equations of Mathematical Physics, 2nd ed.; Higher Education Press: Beijing, China, 2002; pp. 80–86, ISBN 7040107015. (In Chinese) LopezMolina, C.; Baets De, B.; Bustince, H. Quantitative error measures for edge detection. Pattern Recognit. 2013, 46, 1125–1139. [CrossRef] Baddeley, A.J. An error metric for binary images. In Proceedings of the IEEE Workshop on Robust Computer Vision, Bonn, Germany, 9–11 March 1992; Wichmann Verlag: Karlsruhe, Germany, 1992; pp. 59–78. FernándezGarcía, N.L.; MedinaCarnicer, R.; CarmonaPoyato, A.; MadridCuevas, F.J.; PrietoVillegas, M. Characterization of empirical discrepancy evaluation measures. Pattern Recognit. Lett. 2004, 25, 35–47. [CrossRef] Heath, M.D.; Sarkar, S.; Sanocki, T.A.; Bowyer, K.W. A robust visual method for assessing the relative performance of edge detection algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 1997, 19, 1338–1359. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
185
SS symmetry Article
The Complexity of Some Classes of Pyramid Graphs Created from a Gear Graph JiaBao Liu 1 and Salama Nagy Daoud 2,3, * 1 2 3
*
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China; [email protected] Department of Mathematics, Faculty of Science, Taibah University, AlMadinah 41411, Saudi Arabia Department of Mathematics and Computer Science, Faculty of Science, Menouﬁa University, Shebin El Kom 32511, Egypt Correspondence: [email protected]; Tel.: +966598914649
Received: 8 November 2018; Accepted: 23 November 2018; Published: 2 December 2018
Abstract: The methods of measuring the complexity (spanning trees) in a ﬁnite graph, a problem related to various areas of mathematics and physics, have been inspected by many mathematicians and physicists. In this work, we deﬁned some classes of pyramid graphs created by a gear graph then we developed the Kirchhoff’s matrix tree theorem method to produce explicit formulas for the complexity of these graphs, using linear algebra, matrix analysis techniques, and employing knowledge of Chebyshev polynomials. Finally, we gave some numerical results for the number of spanning trees of the studied graphs. Keywords: complexity; Chebyshev polynomials; gear graph; pyramid graphs MSC: 05C05, 05C50
1. Introduction The graph theory is a theory that combines computer science and mathematics, which can solve considerable problems in several ﬁelds (telecom, social network, molecules, computer network, genetics, etc.) by designing graphs and facilitating them through idealistic cases such as the spanning trees, see [1–10]. A spanning tree of a ﬁnite connected graph G is a maximal subset of the edges that contains no cycle, or equivalently a minimal subset of the edges that connects all the vertices. The history of enumerating the number of spanning trees τ ( G ) of a graph G dates back to 1842 when the physicist Kirchhoff [11] offered the matrix tree theorem established on the determinants of a certain matrix gained from the Laplacian matrix L deﬁned by the difference between the degree matrix D and adjacency matrix A, where D is a diagonal matrix, D = dig (d1 , d2 , . . . , dn ) corresponding to a graph G with n vertices that has the vertex degree of di in the ith position of a graph G and A is a matrix with rows and columns labeled by graph vertices, with a 1 or 0 in position (ui , u j ) according to whether ui and u j are adjacent or not. That is ⎧ ⎪ if i = j ⎨ ai Li j = −1 if i = j and i is adjacent to j , ⎪ ⎩ 0 otherwise where ai denotes the degree of the vertex i. This method allows beneﬁcial results for a graph comprising a small number of vertices, but is not feasible for large graphs. There is one more method for calculating τ ( G ). Let λ1 ≥ λ2 ≥ . . . ≥ λk = 0
Symmetry 2018, 10, 689; doi:10.3390/sym10120689
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Symmetry 2018, 10, 689
denote the eigenvalues of the matrix L of a graph G with n vertices. “Kelmans” and “Chelnokov” [12] have derived that 1 k −1 τ ( G ) = ∏ λk . k i =1 One of the favorite methods of calculating the complexity is the contraction–deletion theorem. For any graph G, the complexity τ ( G ) of G is equal to τ ( G ) = τ ( G − e) + τ ( G/e), where e is any edge of G, and where G − e is the deletion of e from G, and G/e is the contraction of e in G. This gives a recursive method to calculate the complexity of a graph [13,14]. Another important method is using electrically equivalent transformations of networks. Yilun Shang [15] derived a closedform formula for the enumeration of spanning trees the subdividedline graph of a simple connected graph using the theory of electrical networks. Many works have conceived techniques to derive the number of spanning trees of a graph, some of which can be found at [16–18]. Now, we give the following Lemma: Lemma 1 [19]. τ ( G ) = k12 det (k I − D c + Ac ) where Ac and D c are the adjacency and degree matrices of G c , the complement of G, respectively, and I is the k × k identity matrix. The characteristic of this formula is to express τ ( G ) straightway as a determinant rather than in terms of cofactors as in Kirchhoff theorem or eigenvalues as in Kelmans and Chelnokov formula. 2. Chebyshev Polynomial In this part we insert some relations regarding Chebyshev polynomials of the ﬁrst and second types which we use in our calculations. We start from their deﬁnitions, see Yuanping, et al. [20]. Let An ( x ) be n × n matrix such that ⎛
2x ⎜ ⎜ −1 ⎜ ⎜ An ( x ) = ⎜ ⎜ 0 ⎜ . ⎜ . ⎝ . 0
−1 2x .. . .. . ···
··· 0 .. . −1 . . . .. .. . . 0 .. .. . . −1 0 −1 2x 0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠
Furthermore, we render that the Chebyshev polynomials of the ﬁrst type are deﬁned by Tn ( x ) = cos(n cos−1 x )
(1)
The Chebyshev polynomials of the second type are deﬁned by Un−1 ( x ) =
1 d sin (n cos−1 x ) Tn ( x ) = n dx sin (cos−1 x )
(2)
It is easily conﬁrmed that Un ( x ) − 2xUn−1 ( x ) + Un−2 ( x ) = 0
(3)
It can then be shown from this recursion that by expanding detAn ( x ) one obtains Un ( x ) = det( An ( x )), n ≥ 1 Moreover, by solving the recursion (3), one gets the straightforward formula
187
(4)
Symmetry 2018, 10, 689
(x +
Un ( x ) =
√
x 2 − 1)
n +1
− (x − 2 x2 − 1
√
√
x 2 − 1)
n +1
, n ≥ 1,
(5)
where the conformity is valid for all complex x (except at x = ±1, where the function can be taken as the limit). The deﬁnition of Un ( x ) easily yields its zeros and it can therefore be conﬁrmed that n −1
Un−1 ( x ) = 2n−1 ∏ ( x − cos j =1
One further notes that
jπ ) n
Un−1 (− x ) = (−1)n−1 Un−1 ( x )
(6)
(7)
From Equations (6) and (7), we have: n −1
Un−1 2 ( x ) = 4n−1 ∏ ( x2 − cos2 j =1
jπ ) n
(8)
Finally, straightforward manipulation of the above formula produces the following formula (9), which is highly beneﬁcial to us later: % Un−1 ( 2
x+2 )= 4
n −1
∏ (x − 2 cos j =1
2jπ ) n
(9)
Moreover, one can see that Un−1 2 ( x ) = Tn ( x ) =
1 − T2n ( x ) 1 − Tn (2x2 − 1) = 2 2(1 − x ) 2(1 − x 2 )
n n 1 [( x + x2 − 1 ) + (( x − x2 − 1 ) ] 2
(10)
(11)
Now we introduce the following important two Lemmas. Lemma 2 [21]. Let Bn ( x ) be n × n Circulant matrix such that ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Bn ( x ) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
x 0 1 .. . 1 0
0 .. . .. . .. . .. . 1
1 .. . .. . .. . .. . ···
··· .. . .. . .. . .. . 1
1 .. . .. . .. . .. . 0
0
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ 1 ⎟ ⎟ ⎟ 0 ⎠ x
1 .. .
Then for n ≥ 3, x ≥ 4, one has det( Bn ( x )) =
2( x + n − 3) x−1 [ Tn ( ) − 1]. x−3 2
Lemma 3 [22]. If A ∈ F n×n , B ∈ F n×m , C ∈ F m×n and D ∈ F m×m . Suppose that A and D are nonsingular matrices, then: det
A C
B D
= det( A − BD −1 C )detD = detAdet( D − CA−1 B).
This Lemma gives a type of symmetry for some matrices which simplify our calculations of the complexity of graphs studied in this paper. 188
Symmetry 2018, 10, 689
3. Main Results (m)
is the graph created from the gear graph Deﬁnition 1. The pyramid graph An Gm+1 with vertices {u0 ; u1 , u2 , . . . , um ; w1 , w2 , . . . , wm } and m sets of vertices, say, 3 1 1 4 3 4 3 4 v1 , v2 , . . . , v1n , v21 , v22 , . . . , v2n , . . . , v1m , v2m , . . . , vm n , such that for all i = 1, 2, . . . , n the vertex j
vi is adjacent to u j and u j+1 ,where j = 1, 2, . . . , m − 1, and vim is adjacent to u1 and um . See Figure 1. X
Z
Y
Y
Y
Y Q
Y
Q
Y
Y
Z
Y
X
Y Q Y Y Y X
Z
X
(3)
Figure 1. The pyramid graph An . (m)
Theorem 1. For n ≥ 0, m ≥ 3 , τ ( An ) = 2mn [(n + 2 + 2( n + 1) m ] .
√
m
2n + 3 ) + (n + 2 −
√
m
2n + 3 ) −
Proof. Using Lemma 1, we have (m)
τ ( An ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ det ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 0 ( m + 1) 0 2( n + 2) 1 .. .. . . 1 .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· 0 1 1 0 0 .. .. . . 1 .. .. .. . . . .. .. .. . . . .. .. . . 1 1 0 1 1 0 0 .. .. .. . . . .. . 0 0 .. . 1 0 .. .. .. . . . .. . 1 0 .. . 1 1 .. .. .. . . . .. . 1 1 .. .. .. . . . .. .. .. . . . .. . 1 1 .. .. . . . .. .. . 1 1 .. . 0 1 .. .. .. . . . 1 0 1
1 (mn+2m+1)2
··· ··· ··· 0 ··· ··· ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. . . . 1 · · · · · · 1 2( n + 2) 1 1 ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. . . . 1 .. .. .. . . . 0 ··· ··· 1 0 0 1 ··· 1 .. .. .. .. . . . . .. .. . . 1 1 .. .. 0 . 1 . .. ... ... ... . .. .. 0 . 1 . . . .. . . 0 . . . .. .. .. .. . . . . . .. 0 .. 1 . . . .. 1 .. .. . . . .. .. .. 1 . .. .. . . 0 1 .. .. .. .. . . . . .. .. .. . . 0 . .. .. 1 . 0 . .. .. .. .. . . . . 1
···
0
···
× det((mn + 2m + 1) I − D c + Ac ) =
··· ··· 1 ··· .. .. . . 0 .. .. . . 1 .. . . .. . . . ... . . . . . . 1 ··· ··· 3 1 ··· .. .. . . 1 . .. .. . . .. .. . . .. . . . .. . . .. . . . 1 ··· ··· 1 1 ··· .. .. .. . . . 1 0
··· ··· ··· 1 .. .. . . .. .. . . .. .. . . .. .. . . 1 0 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· 1 .. .. . .
1
1
··· ···
1
1 .. .
1 .. .
··· ··· .. ... .
1 .. .
1 .. . .. . .. . .. . .. . .. . .. .
1 .. . .. . .. . .. . .. . .. . .. .
··· ··· .. ... . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . .
1 .. . .. . .. . .. . .. . .. . .. .
1
1
··· ···
1
1 .. . 1
1 .. . 1
··· ··· .. .. . . ··· ···
1 .. . 1
1 × (mn+2m+1)2
1 1 0 0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 0 1 ··· 1 ··· ··· 1 ··· 1 1 ··· 1 0 ···
1 0 .. . 1 .. . 1 . 1 .. 0 1 1 1 .. . 1 . . .. .. .. .. . . .. . 1 3 1 1 3 .. . 1 . 1 .. . 1 .. ... ... . 1 .. ... ... ... ... . . .. .. . . .. .. . . .. .. .. .. . . .. .. . . . 1 .. . 1 .. .. .. . . 1 1
···
0
0
···
0
1
···
1
···
1
0
···
0
0
···
0
···
1
1
···
1
0
···
0
189
··· ···
1
···
1
···
1
···
1
··· ···
1
···
0
···
0
···
1
···
1
1
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 1 1 ··· 1 ··· ··· 1 ··· 1 0 ··· 0 0 ··· ··· 1 1 ··· 1 ··· ··· ··· ··· ··· 1 ··· 1 1 ··· ··· ··· ···
1 .. . .. .
1 .. . .. .
··· 1 1 ··· 1 1 1 ··· ··· .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· ···
··· ··· ···
1 .. . .. .
··· 1 ··· 1 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
1 .. . .. .
··· 1 ··· 1 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ···
1 .. . .. .
1 .. . .. .
··· ··· ···
··· 1 1 ··· ··· 1 1 ··· ··· ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1
⎞ 1 ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ ··· ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ 1 ⎟ ⎟ ... ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ .. ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ ... ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ . ⎟ ⎟ .. ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ ⎟ 1 ⎠ 3
Symmetry 2018, 10, 689
Let j = (1 · · · 1) be the 1 × n matrix with all one, and Jn be the n × n matrix with all one. Set a = 2n + 4 and b = mn + 2m + 1. Then we obtain: ⎛
(m) τ An =
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎛
=
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b b .. . .. . .. . .. . b b .. . .. . .. . .. . b bjt .. . .. . .. . .. . bjt 1 1 .. . .. . .. . .. . 1 1 .. . .. . .. . .. . 1 1jt .. . .. . .. . .. . 1jt
m+1 0 .. . .. . .. . .. . 0 1 .. . .. . .. . .. . 1 jt .. . .. . .. . .. . jt 0 a 1 .. . .. . .. . 1 0 1 .. . .. . 1 0 0 jt .. . .. . jt 0 0 a 1 .. . .. . .. . 1 0 1 .. . .. . 1 0 0 jt .. . .. . jt 0
0 a 1 .. . .. . .. . 1 0 1 .. . .. . 1 0 0 jt .. . .. . jt 0
··· ··· 1 ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· ··· 0 1 .. . 0 ..
.
..
.
..
. 1 0
0 .. . .. . .. . t j
··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· ··· .. .. . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· 1 jt · · · · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . t ··· ··· j
··· ··· 1 ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· ··· 0 1 .. . 0 ..
.
..
.
..
. 1 0
0 .. . .. . .. . jt
··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· ··· .. .. . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· 1 jt · · · · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . · · · · · · jt
··· ··· 1 ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· ··· 0 1 .. . 0 ..
.
..
.
..
. 1 0
0 .. . .. . .. . jt
··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· ··· .. .. . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· 1 jt · · · · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . · · · · · · jt
0 1 .. . .. . .. . 1 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0 0 1 .. . .. . .. . 1 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0
1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt 1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
0 1 .. . .. . .. .
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . .
1 0 0 1 .. . .. . 1 3
1 a 1 1 .. .
1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
1 0 0 jt .. . .. . jt 0 0
.. ..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
190
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt 1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
··· j .. . 0 .. . j .. . . . . ... . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· j .. . 0 .. . j .. . . . . ... . . . j ··· j ··· .. . . . . .. . . . . ... . . . .. . . . .
··· ··· .. . .. . .. . .. . ··· ··· .. .
j 0
j
..
.
..
.
..
. ···
···
2Im n + Jm n
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ . ⎟ .. .. . . . . ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0
⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
··· ··· 1 ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· ··· 0 1 .. . 0
0 a 1 .. . .. . .. . 1 0 1 .. . .. .
1 0 0 jt .. . .. . jt 0
.
0 .. . .. . .. . jt
jt 0
1 .. . .. .
..
. 1 0
jt ... .. .
a
.
..
1 0 0
1 .. . .. . .. . 1 0
..
1 .. . .. . .. . .. . ··· 0 0 .. . .. . .. . 1 0 0 .. . .. . .. . jt
··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· ··· .. .. . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· 1 jt · · · · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . · · · · · · jt
··· ··· ··· .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· 1 1 ··· ··· .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· 1 jt · · · · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . · · · · · · jt
··· ··· ··· 1 j ··· ··· · · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . −1 −1 0 −j . .. .. .. .. .. .. .. . . . . . . 0 0 .. .. .. .. .. .. .. .. ... . . . . . . . . .. .. .. .. .. .. .. . . . −1 0 . . . . 0 · · · · · · 0 −1 −1 0 · · · · · · 2 0 ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. .. . . . . . . . 0 . .. .. .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. .. .. . . . . . . . . 0 0 ··· ··· ··· 0 2 0 ··· ··· 0 ··· ··· ··· ··· 0 .. . .. .. .. .. .. . . . . . .. .. .. .. .. ... . . . . . 2Im n .. .. .. .. .. .. . . . . . . .. .. .. .. .. ... . . . . . 0 ··· ··· ··· ··· 0
0 1 .. . .. . .. .
··· 0
1 −1
1 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0
−1
1 .. . .. . .. .
0
−1 −1 .. . 0 .. .. . . .. ... . 0 ··· 2 0 .. . 0 .. .. . . .. .. . . .. ... . 0 ··· 0 ··· .. .. . . .. .. . . .. ... . .. .. . .
1 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0
···
0
· · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . 0 −j .. .. .. .. .. .. . . . . . . 0 .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. . . −1 0 . . . · · · 0 −1 −1 0 · · · · · · ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 0 ··· ··· 0 2 0 ··· ··· ··· ··· ··· 0 .. .. .. .. . . . . .. .. .. .. . . . . 2Im n .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 0
⎞ ··· ··· j · · · 0 −j ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ · · · 0 −j ⎟ .. .. . . 0 ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. .. ... ⎟ ⎟ . . ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. ... ⎟ . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 3, yields (m)
τ ( An ) =
1 b
⎛
2a ⎜ ⎜ n+2 ⎜ ⎜ ⎜ 2( n + 1) ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎜ ⎜ 2( n + 1) ⎜ ⎜ n+2 = 1b 2mn × 2−2m × det⎜ ⎜ 0 ⎜ ⎜ ⎜ 2 ⎜ ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎝ 2 0
× det
A C
B 2Imn
=
1 b
× det( A − B 2I1mn C ) × 2mn
n+2
2( n + 1)
···
2( n + 1)
n+2
−2
2a
n+2 .. . .. . .. . ··· 2 .. . .. . .. . .. . ···
2( n + 1) .. . .. . .. . 2( n + 1) ··· .. . .. . .. . .. . ···
··· .. . .. . .. . n+2 ··· .. .
2( n + 1) .. .
−2
n+2 .. . .. . 2( n + 1) 0 .. . .. . .. . .. . 2
2( n + 1)
..
.
n+2 2a 2 .. . .. .
..
.
2
. 2
0 0
..
0 .. . .. . 0 4 0 .. . .. . .. . 0
0 .. . .. . .. . .. . ··· 0 .. . .. . .. . .. . ···
⎞ · · · · · · 0 −2 ⎟ .. .. .. . . . 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. .. .. ⎟ . . . 0 ⎟ ⎟ · · · 0 −2 −2 ⎟ ⎟ ··· ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ ⎟ .. .. .. . . . 0 ⎠ ··· ··· 0 4
Using Lemma 3 again, yields (m) τ ( An )
2m n−2m = × det b
D F
E 4Im
191
=
1 2m n × det( D − E F) b 4Im
Symmetry 2018, 10, 689 ⎛
(m)
τ ( An ) =
2mn b
2a ⎜ ⎜ ( n + 3) ⎜ ⎜ ⎜ 2( n + 2) ⎜ × det⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎝ 2( n + 2) ( n + 3)
( n + 3)
2( n + 2)
2a .. . .. . .. . 2( n + 2)
( n + 3) .. . ..
⎞
··· 2( n + 2) ( n + 3) .. . ··· 2( n + 2) .. .. .. . . . .. .. . . 2( n + 2) .. .. . . ( n + 3) 2( n + 2) ( n + 3) 2a
.
..
. ···
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Straightforward inducement using the properties of determinants, one can obtain ⎛
(m)
τ ( An ) =
2mn b
×
2b m n + m +2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛
2m n +1 ( n +1) m m n + m +2
=
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(2a − n − 3)
0
( n + 1)
0
(2a − n − 3) .. .
0 .. . .. . .. . ···
( n + 1) .. .
..
.
..
( n + 1) 0
. ( n + 1)
(2a−n−3) ( n +1)
0
0
(2a−n−3) ( n +1)
1
1
..
.
0 .. . .. . .. .
0
1
···
1 .. .
..
.
..
.
··· ( n + 1) 0 .. . ··· ( n + 1) .. .. .. . . . .. .. . . ( n + 1) .. .. . . 0 ( n + 1) 0 (2a − n − 3)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎞
··· .. . .. . .. . .. .
1 .. .
1
0
0
..
.
1 .. .
..
.
1
..
.
0
(2a−n−3) ( n +1)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 2, yields (m)
2( 2an−+n1−3 +m−3) ( n +1) m × [ Tm ( 2a−n−3 −3 mn+m+2 × n +1 m n +2 mn + 1 2 × (n + 1) × [ Tm ( n+1 ) − 1].
τ ( A n ) = 2m n +1 ×
=
2a−n−3 −1 n +1
2
) − 1]
Using Equation (11), yields the result. (m)
Deﬁnition 2. The pyramid graph Bn is the graph created from the gear graph Gm+1 with vertices {u0 ; u1 , u2 , . . . , um ; w1 , w2 , . . . , wm } with double internal edges and m sets of vertices, say, 3 1 1 4 3 4 3 4 j v1 , v2 , . . . , v1n , v21 , v22 , . . . , v2n , . . . , v1m , v2m , . . . , vm n , such that for all i = 1, 2, . . . , n the vertex vi is m adjacent to u j and u j+1 , where j = 1, 2, . . . , m − 1, and vi is adjacent to u1 and um . See Figure 2. X
Z
Y
Y
Y Y
Y
Q
Y
Y
Z
Y Q X
Y Q Y Y Y X
X
Z
(3)
Figure 2. The pyramid graph Bn . 192
Symmetry 2018, 10, 689
√ √ m m (m) Theorem 2. For n ≥ 0, m ≥ 3, τ ( Bn ) = 2mn [(n + 3 + 2 n + 2 ) + (n + 3 − 2 n + 2 ) − m 2( n + 1) ] . Proof. Using Lemma 1, we get: (m)
τ ( Bn ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ det ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
−1 −1 (2m + 1) −1 (2n + 5) 1 .. .. . . 1 .. .. .. . . . .. .. .. . . . .. .. .. . . . −1 1 ··· 1 0 0 .. .. . . 1 .. .. .. . . . .. .. .. . . . .. .. . . 1 1 0 1 1 0 0 .. .. .. . . . .. . 0 0 .. . 1 0 .. .. .. . . . .. . 1 0 .. . 1 1 .. .. .. . . . .. . 1 1 .. .. .. . . . .. .. .. . . . .. . 1 1 .. .. .. . . . .. . 1 1 .. . 0 1 .. .. .. . . . 1 0 1
1 (mn+2m+1)2
··· ··· ··· −1 ··· ··· ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. . . . 1 · · · · · · 1 (2n + 5) 1 ··· ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. . . . 1 .. .. .. . . . 0 ··· ··· 1 0 1 ··· 1 0 .. .. .. .. . . . . .. .. 1 . 1 . .. .. 0 . 1 . .. ... ... ... . .. .. 0 . 1 . . . .. . . 0 . . . .. .. .. .. . . . . .. .. 0 . 1 . . . .. 1 .. .. . . . .. .. .. 1 . . .. 1 .. 0 . .. .. .. .. . . . . .. .. .. . . 0 . .. .. 1 . 0 . .. .. .. .. . . . . 1
···
0
···
× det((mn + 2m + 1) I − D c + Ac ) =
··· ··· 1 ··· .. .. . . 0 .. .. . . 1 .. . . .. . . . ... . . . . . . 1 ··· ··· 3 1 ··· .. .. . . 1 . .. .. . . .. . .. .. . . .. . .. .. . . .. 1 ··· ··· 1 1 ··· .. .. .. . . . 1 0
··· ··· ··· 1 .. .. . . .. .. . . .. .. . . .. .. . . 1 0 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· 1 .. .. . .
1
1
··· ···
1
1 .. .
1 .. .
··· ··· .. .. . .
1 .. .
1 .. . .. . .. . .. . .. . .. . .. .
1 .. . .. . .. . .. . .. . .. . .. .
··· ··· .. ... . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . .
1 .. . .. . .. . .. . .. . .. . .. .
1
1
··· ···
1
1 .. . 1
1 .. . 1
··· ··· .. .. . . ··· ···
1 .. . 1
1 × (mn+2m+1)2
1 1 0 0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 0 1 ··· 1 ··· ··· 1 ··· 1 1 ··· 1 0 ···
1 0 .. . 1 .. . 1 . 1 .. 0 1 1 1 .. . 1 . . .. .. . . .. .. .. . 1 3 1 1 3 .. . 1 . 1 .. . 1 .. ... ... . 1 .. ... ... ... ... ... ... ... ... . . .. .. . . .. .. .. .. . . . 1 .. . 1 .. .. .. . . 1 1
···
0
0
···
0
1
···
1
···
1
0
···
0
0
···
0
···
1
1
···
1
0
···
0
··· ···
1
···
1
···
1
···
1
··· ···
1
···
0
···
0
···
1
···
1
1
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 1 1 ··· 1 ··· ··· 1 ··· 1 0 ··· 0 0 ··· ··· 1 1 ··· 1 ··· ··· ··· ··· ··· 1 ··· 1 1 ··· ··· ··· ···
1 .. . .. .
···
1 .. . .. .
··· ···
1 .. . .. .
··· 1 1 ··· 1 ··· 1 1 ··· 1 1 ··· ··· ··· ··· .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . ··· ··· ··· ··· ···
··· ··· ··· ··· ···
1 .. . .. .
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ···
1 .. . .. .
1 .. . .. .
··· ··· ···
··· 1 ··· 1 1 ··· ··· 1 ··· 1 1 ··· ··· ··· ··· ··· ··· ··· .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . ··· ··· ··· ··· ··· 1
⎞ 1 ⎟ 0 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ ··· ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ . ⎟ .. ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ 1 ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ ... ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ . ⎟ ⎟ .. ⎟ ⎟ .. ⎟ . ⎟ ⎟ . ⎟ .. ⎟ ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ ⎟ 1 ⎠ 3
Let j = (1 · · · 1) be the 1 × n matrix with all one, and Jn be the n × n matrix with all one. Set a = 2n + 5 and b = mn + 2m + 1. Then we get: ⎛
(m) τ Bn =
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
2m + 1 −1 .. . .. . .. . .. . −1 1 .. . .. . .. . .. . 1 jt .. . .. . .. . .. . jt
−1 · · · · · · a 1 ··· .. .. . . 1 .. .. .. . . . .. .. .. . . . .. .. .. . . . 1 ··· ··· 0 0 1 .. . 1 0 .. .. .. . . . .. .. .. . . . .. .. . . 1 0 1 ··· 0 0 jt .. . jt 0 .. .. .. . . . .. .. .. . . . .. .. . . jt 0 jt · · ·
· · · · · · −1 ··· ··· 1 .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. . . 1 ··· 1 a ··· ··· 1 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. . . 0 ··· 1 0 · · · · · · jt .. .. .. . . . .. .. .. . . . .. .. . . jt .. . ···
..
. jt
0 0
1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
193
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . .
1 0 1 .. . .. .
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
1 0 1 .. . .. . .. .
. 1 ··· .. . .. . .. . .. . ···
1 3 jt .. . .. . .. . .. . jt
..
. ··· ··· .. . .. . .. . .. . ···
..
. ··· ··· .. . .. . .. . .. . ···
..
. ··· ··· .. . .. . .. . .. . ···
..
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ . ⎟ .. .. ⎟ . . .. ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0
b b .. . .. . .. . .. . b b .. . .. . .. . .. . b bjt .. . .. . .. . .. . bjt
−1 · · · · · · a 1 ··· .. .. . . 1 .. .. .. . . . .. .. ... . . .. .. .. . . . 1 ··· ··· 0 0 1 .. . 1 0 .. .. .. . . . .. .. .. . . . .. .. . . 1 0 1 ··· 0 0 jt .. . jt 0 .. .. .. . . . .. .. .. . . . .. .. t . . j 0 jt · · ·
1 1 .. . .. . .. . .. . 1 1 .. . .. . .. . .. . 1 1jt .. . .. . .. . .. . 1jt
−1 · · · · · · a 1 ··· .. .. . . 1 .. .. .. . . . .. .. ... . . .. .. .. . . . 1 ··· ··· 0 0 1 .. . 1 0 .. .. .. . . . .. .. .. . . . 1 0 0 jt ... .. . jt 0
..
. 1 0
0 .. . .. . .. . jt
· · · · · · −1 ··· ··· 1 .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. . . 1 ··· 1 a ··· ··· 1 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. . . 0 ··· 1 0 · · · · · · jt .. .. .. . . . . .. .. . . . . .. .. . . jt .. . ···
..
. jt
0 0
· · · · · · −1 ··· ··· 1 .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. . . 1 ··· 1 a ··· ··· 1 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. .. . . . 0 ··· ··· 1 0 t j · · · · · · jt .. .. .. .. . . . . . .. .. .. . . . . . .. .. .. . . . jt .. .. . . ··· ···
..
. jt
0 0
1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt 1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
−1 ··· ··· ( a + 1) 2 · · · .. .. . . 2 .. .. .. . . . .. .. ... . . .. .. .. . . . 2 ··· ··· 1 1 2 .. . 2 1 .. .. .. . . . .. .. .. . . . 2 1 jt 2jt ... .. . 2jt jt
..
. 2 t j
jt .. . .. ..
.
. 2jt
··· ··· −1 1 ··· ··· ··· 2 −1 0 .. .. .. . . . −1 −1 .. .. .. .. . . . . 0 .. .. .. .. .. . . . . . .. .. .. .. . . . . 2 · · · 2 ( a + 1) 0 · · · ··· ··· 2 2 0 .. .. .. . . . 2 0 .. .. .. .. .. . . . . . .. .. .. .. . . . . 2 .. .. .. .. .. . . . . . 1 ··· ··· 2 1 0 ··· 2jt · · · · · · 2jt 0 ··· .. .. .. .. .. .. . . . . . . .. . .. .. .. .. . . . . . . . .. .. .. .. .. . . . . 2jt . . .. .. .. .. . t . . . . . j · · · · · · 2jt jt 0 ···
194
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt 1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . .. . . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· ··· ··· 1 j ··· ··· · · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . 0 −j .. .. .. .. .. .. . . . . . . 0 .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. . . −1 0 . . . · · · 0 −1 −1 0 · · · · · · ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 0 ··· ··· 0 2 0 ··· ··· ··· ··· ··· 0 .. .. .. .. . . . . .. .. .. .. . . . . 2Im n .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 0
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . j ⎟ . ⎟ ⎟ . .. .. ⎟ . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ ⎟ . .. .. ⎟ . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ . .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ··· ··· j · · · 0 −j ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
( a + 1)
··· ··· ··· 2 −1 0 .. .. .. .. . . . . −1 −1 .. .. .. .. .. . . . . . 0 .. .. .. .. .. .. . . . . . . .. .. .. .. .. . . . . . 2 · · · · · · 2 ( a + 1) 0 · · · 2 ··· ··· 2 2 0 .. .. .. .. . . . . 2 0 .. .. .. .. .. .. . . . . . . .. .. .. .. .. . . . . . 2 .. .. .. .. .. . . . . . 1 ··· ··· 2 1 0 ··· 2jt · · · · · · 2jt 0 ··· .. .. .. .. .. .. . . . . . . .. .. .. .. .. .. . . . . . . .. .. .. .. .. . . . . 2jt . .. .. .. .. .. . . . . . jt · · · · · · 2jt jt 0 ···
2 .. . .. . .. . .. . ··· 1
2 .. . .. . .. . 2 1 2 .. . .. .
1 .. . .. . .. . 2 jt
2 1 jt 2jt .. . .. .
jt .. . ..
.
..
2jt jt
. 2jt
· · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . 0 −j .. .. .. .. .. .. . . . . . . 0 .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. . . −1 0 . . . · · · 0 −1 −1 0 · · · · · · ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 0 ··· ··· 0 2 0 ··· ··· ··· ··· ··· 0 .. .. .. .. . . . . .. .. .. .. . . . . 2Im n .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 0
⎞ · · · 0 −j ⎟ .. .. . 0 ⎟ . ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. .. ... ⎟ ⎟ . . ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ . ⎟ .. .. . . . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 3, yields (m) τ ( Bn ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 mn − 2m ⎜ = b2 × 2 × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
=
1 b
A C
× det
B 2Imn
=
1 b
× det( A − B 2I1mn C ) × 2mn
(2a + 2n + 2)
3n + 4
4( n + 1)
···
4( n + 1)
3n + 4
−2
3n + 4
(2a + 2n + 2) 3n + 4 .. . .. . 4( n + 1) 2 .. . .. . .. . .. . 4
4( n + 1) .. . .. . .. . 4( n + 1) ··· .. . .. . .. . .. . ···
··· .. . .. . .. . 3n + 4 ··· .. .
4( n + 1) .. .
−2
4( n + 1) .. .
3n + 4 .. . .. . .. . ··· 4 .. . .. . .. . .. . ···
4( n + 1) 3n + 4 2 4 .. . .. . 4 2
4( n + 1)
..
.
3n + 4 (2a + 2n + 2) 4 .. . .. .
..
.
4
. 4
2 2
..
0 .. . .. . 0 4 0 .. . .. . .. . 0
0 .. . .. . .. . .. . ··· 0 .. . .. . .. . .. . ···
⎞ · · · · · · 0 −2 ⎟ .. .. .. . . . 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. .. .. ⎟ . . . 0 ⎟ ⎟ · · · 0 −2 −2 ⎟ ⎟ ··· ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ ⎟ .. .. .. . . . 0 ⎠ ··· ··· 0 4
Using Lemma 3 again, yields (m)
τ ( Bn ) = ⎛
(m)
τ ( Bn ) =
2mn b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎝
2m n−2m b
× det
(2a + 2n + 4) (3n + 7) 4( n + 2) .. . 4( n + 2) (3n + 7)
D F
E 4Im
(3n + 7)
=
2m n b
× det( D − E 4I1m F )
4( n + 2)
··· 4( n + 2) (3n + 7) .. . (2a + 2n + 4) (3n + 7) ··· 4( n + 2) .. .. .. .. .. . . . . . .. .. .. .. . . . . 4( n + 2) .. .. .. .. . . . . (3n + 7) 4( n + 2) ··· 4(n + 2) (3n + 7) (2a + 2n + 4)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
With a straightforward inducement using properties of determinants, we obtain ⎛
(m)
τ ( Bn ) =
2mn b
×
4b m n + m +4
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(2a − n − 3)
0
( n + 1)
0
(2a − n − 3) .. .
0 .. . .. . .. . ···
( n + 1) .. . ( n + 1) 0
.. ..
.
. ( n + 1)
195
··· ( n + 1) 0 .. . ··· ( n + 1) .. .. .. . . . .. .. . . ( n + 1) .. .. . . 0 ( n + 1) 0 (2a − n − 3)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
2m n +2 × ( n +1) m m n + m +4
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(2a−n−3) ( n +1)
0
0
(2a−n−3) ( n +1)
1
..
.
..
.
1
..
.
0 .. . .. . .. .
0
1
···
1 .. .
⎞
··· .. . .. . .. . .. .
1 .. .
1
0
0
..
.
1 .. .
..
.
1
..
.
0
(2a−n−3) ( n +1)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 2, yields (m)
2( 2an−+n1−3 +m−3) ( n +1) m × [ Tm ( 2a−n−3 −3 mn+m+4 × n +1 3 2mn+1 × (n + 1)m × [ Tm ( nn+ +1 ) − 1] .
τ ( Bn ) = 2m n+2 ×
=
2a−n−3 −1 n +1
2
) − 1]
Using Equation (11), yields the result. (m)
Deﬁnition 3. The pyramid graph Cn is the graph created from the gear graph Gm+1 with vertices {u0 ; u1 , u2 , . . . , um ; w1 , w2 , . . . , wm } with double external edges and m sets of vertices, say, 3 1 1 4 3 4 3 4 j v1 , v2 , . . . , v1n , v21 , u22 , . . . , v2n , . . . , v1m , v2m , . . . , vm n , such that for all i = 1, 2, . . . , n the vertex vi is m adjacent to u j and u j+1 , where j = 1, 2, . . . , m − 1, and vi is adjacent to u1 and um . See Figure 3. X
Z
Y
Y Y Y
Q
Y Q
Y
Y
Y
Z
X
Y Q Y Y Y X
Z
X
(3)
Figure 3. The pyramid graph Cn . (m)
Theorem 3. For n ≥ 0, m ≥ 3, τ (Cn ) = 2mn [(n + 4 + 2( n + 3) m ].
√
m
2n + 7 ) + (n + 4 −
Proof. Using Lemma 1, we have: (m)
τ (Cn ) =
1 (mn+2m+1)2
× det((mn + 2m + 1) I − D c + Ac ) =
196
1 × (mn+2m+1)2
√
m
2n + 7 ) −
Symmetry 2018, 10, 689 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ det ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 ( m + 1) 0 2( n + 3) .. . 0 .. . 1 .. .. . . .. . 1 0 0 1 0 .. . 1 .. .. . . .. .. . . .. . 1 1 0 1 0 .. .. . . .. . 0 .. . 1 .. .. . . .. . 1 .. . 1 .. .. . . .. . 1 .. .. . . .. .. . . .. . 1 .. .. . . .. . 1 .. . 0 .. .. . . 1
0
0 0 .. . .. . .. . .. . 1 0 .. . .. . .. . .. . 1 0 .. . 0 0 .. . 0 1 .. . 1 .. . .. . 1 .. . 1 1 .. . 1
··· ··· ··· 0 1 ··· 1 0 .. .. .. . . . 1 .. .. .. .. . . . . .. .. .. . . . 1 .. .. .. . . . 0 ··· 1 0 2( n + 3) 1 ··· ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. . . . 1 .. .. .. . . . 0 ··· ··· 1 0 0 1 ··· 1 .. .. .. .. . . . . .. .. 1 . 1 . .. .. 0 . 1 . .. ... ... ... . .. .. 0 . 1 . . . .. . . 0 . . . .. .. .. .. . . . . . .. 0 .. 1 . . . .. 1 .. .. . . . .. .. .. 1 . .. .. 1 . 0 . .. .. .. .. . . . . .. .. .. . . 0 . .. .. 1 . 0 . .. .. .. .. . . . . 1
···
···
0
··· ··· 1 ··· .. .. . . 0 .. .. . . 1 .. . . .. . . . ... . . . . . . 1 ··· ··· 3 1 ··· .. .. . . 1 . .. .. . . .. .. . . .. . . . .. . . .. . . . 1 ··· ··· 1 1 ··· .. .. .. . . . 1 0
··· ··· ··· 1 .. .. . . .. .. . . .. .. . . .. .. . . 1 0 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1 ··· 1 .. .. . .
1
1
··· ···
1
1 .. .
1 .. .
··· ··· .. .. . .
1 .. .
1 .. . .. . .. . .. . .. . .. . .. .
1 .. . .. . .. . .. . .. . .. . .. .
··· ··· .. ... . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . .
1 .. . .. . .. . .. . .. . .. . .. .
1
1
··· ···
1
1 .. . 1
1 .. . 1
··· ··· .. .. . . ··· ···
1 .. . 1
1 1 0 0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 0 1 ··· 1 ··· ··· 1 ··· 1 1 ··· 1 0 ···
1 0 .. . 1 .. . 1 . 1 .. 0 1 1 1 .. . 1 . . .. .. .. .. . . .. . 1 3 1 1 3 .. . 1 . 1 .. . 1 .. ... ... . 1 .. ... ... ... ... . . .. .. . . .. .. . . .. .. .. .. . . .. .. . . . 1 .. . 1 .. .. .. . . 1 1
···
0
0
···
0
1
···
1
···
1
0
···
0
0
···
0
···
1
1
···
1
0
···
0
··· ···
1
···
1
···
1
···
1
··· ···
1
···
0
···
0
···
1
···
1
1
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 1 1 ··· 1 ··· ··· 1 ··· 1 0 ··· 0 0 ··· ··· 1 1 ··· 1 ··· ··· ··· ··· ··· 1 ··· 1 1 ··· ··· ··· ···
1 .. . .. .
···
1 .. . .. .
1 .. . .. .
··· ···
··· 1 1 ··· 1 1 1 ··· ··· .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· ···
··· 1 ··· 1 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ··· ··· ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
1 .. . .. .
··· 1 ··· 1 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . ··· ···
··· ··· ···
1 .. . .. .
1 .. . .. .
··· ··· ···
··· 1 1 ··· ··· 1 1 ··· ··· ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1
⎞ 1 ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ ··· ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ 1 ⎟ ⎟ ... ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ .. ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ ... ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ . ⎟ ⎟ .. ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ ⎟ 1 ⎠ 3
Let j = (1 · · · 1) be the 1 × n matrix with all one, and Jn be the n × n matrix with all one. Set a = 2n + 6 and b = mn + 2m + 1. Then we have: ⎛
(m) τ Cn =
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
m+1 0 .. . .. . .. . .. . 0 1 .. . .. . .. . .. . 1 jt .. . .. . .. . .. . jt
0 a 0 1 .. . 1 0 0 1 .. . .. . 1 0 0 jt ... .. . jt 0
··· ··· ··· ··· 0 1 ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· 1 0 0 1 ··· ··· .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· ··· 1 0 jt · · · · · · .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . jt · · · · · · jt
0 0 1 .. . 1 0 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0
1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
197
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
··· j .. . 0 .. . j .. . . . . .. . . . . ··· ··· .. .
··· ··· .. . .. . .. . .. . ··· ··· .. .
..
.
..
.
..
.
..
.
j 0
j j ... .. . .. . .. . j
..
. ···
..
. ···
2Im n + Jm n
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 b2
⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
b b .. . .. . .. . .. . b b .. . .. . .. . .. . b bjt .. . .. . .. . .. . bjt 1 1 .. . .. . .. . .. . 1 1 .. . .. . .. . .. . 1 1jt .. . .. . .. . .. . 1jt 1 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0
0 a 1 .. . .. . 1 0 0 1 .. . .. . 1 0 0 jt ... .. . jt 0 0 a 0 1 .. . 1 0 0 1 .. . .. . 1 0 0 jt ... .. . jt 0 0 a 1 .. . .. . 1 0 0 1 .. . .. . 1 0 0 jt ... .. . jt 0
··· ··· ··· ··· 0 1 ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· 1 0 0 1 ··· ··· .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· ··· 1 0 jt · · · · · · .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . jt · · · · · · jt ··· ··· ··· ··· 0 1 ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· 1 0 0 1 ··· ··· .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· ··· 1 0 jt · · · · · · .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . jt · · · · · · jt ··· ··· ··· ··· 0 1 ··· 1 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· 1 0 0 1 ··· ··· .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . 1 ··· ··· 1 0 jt · · · · · · .. .. .. . . . 0 .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . jt · · · · · · jt
0 0 1 .. . 1 0 a 1 1 .. .
1 0 0 1 .. . .. . 1 3
0 0
1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
0 0
1 0
1 0 0 jt .. . .. . jt
1 .. . 1 0 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0 0 0 1 .. . 1 0 a 1 1 .. . 1 0 0 jt .. . .. . jt 0 0
0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt 1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . .. . . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· ··· ··· 1 j ··· ··· · · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . −1 −1 0 −j .. .. .. .. .. .. .. . . . . . . 0 . 0 .. .. .. .. .. .. .. .. ... . . . . . . . . .. .. .. .. .. .. .. . . . −1 0 . . . . 0 · · · · · · 0 −1 −1 0 · · · · · · 2 0 ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. .. . . . . . . . 0 . .. .. .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. .. .. . . . . . . . . 0 0 ··· ··· ··· 0 2 0 ··· ··· 0 ··· ··· ··· ··· 0 .. .. .. .. .. .. . . . . . . .. .. .. .. .. ... . . . . . 2Im n .. .. .. .. .. .. . . . . . . .. .. .. .. .. ... . . . . . 0 ··· ··· ··· ··· 0 1 −1
··· 0
1 0
198
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . j ⎟ . ⎟ ⎟ . .. .. ⎟ . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ ⎟ . .. .. ⎟ . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ . .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎞ ··· ··· j · · · 0 −j ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
1 b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
a 0 1 .. . 1 0 0 1 .. . .. . 1 0 0 jt .. . .. . jt 0
0 .. . .. . .. . .. . 1 0
··· 1 .. .. . . .. .. . . .. .. . . .. .. . . 1 0 ··· ··· .. .. . .
1 .. . .. . .. . .. . ··· 1 .. . .. . .. . .. . ··· jt .. . .. . .. . .. . ···
0 .. . .. . .. . 1 0 0 .. . .. . .. . jt
..
.
..
.
..
. ··· ··· .. . .. . .. . .. . ···
0
−1
1 .. .
−1 −1 .. . 0 .. .. . . .. ... . 0 ··· 2 0 .. . 0 .. .. . . .. .. . . .. ... . 0 ··· 0 ··· .. .. . . .. .. . . .. ... . .. .. . .
1 0 a 1 1 .. .
..
.
..
.
1
. 1 ··· .. . .. . .. . .. . jt
0 0 jt .. . .. .
..
jt 0 0
· · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . 0 −j .. .. .. .. .. .. . . . . . . 0 .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. . . −1 0 . . . · · · 0 −1 −1 0 · · · · · · ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 0 ··· ··· 0 2 0 ··· ··· ··· ··· ··· 0 .. .. .. .. . . . . .. .. .. .. . . . . 2Im n .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 0
0
···
0
⎞ · · · 0 −j ⎟ .. .. . 0 ⎟ . ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. .. ... ⎟ ⎟ . . ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ . ⎟ .. .. . . . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 3, yields (m) τ (Cn )
=
1 b
× det
⎛
=
1 b
2a ⎜ ⎜ n ⎜ ⎜ ⎜ 2( n + 1) ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎜ ⎜ 2( n + 1) ⎜ ⎜ n × 2mn × 2−2m × det⎜ ⎜ 0 ⎜ ⎜ ⎜ 2 ⎜ ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎝ 2 0
A C
B 2Imn
=
1 b
× det( A − B 2I1mn C ) × 2mn
n
2( n + 1)
···
2( n + 1)
n
−2
2a
n+2 .. . .. . .. . ··· 2 .. . .. . .. . .. . ···
2( n + 1) .. . .. . .. . 2( n + 1) ··· .. . .. . .. . .. . ···
··· .. . .. . .. . n ··· .. .
2( n + 1) .. .
−2
n .. . .. . 2( n + 1) 0 .. . .. . .. . .. . 2
2( n + 1) n 2a 2 .. . .. .
..
.
..
.
2
. 2
0 0
..
0 .. . .. . 0 4 0 .. . .. . .. . 0
0 .. . .. . .. . .. . ··· 0 .. . .. . .. . .. . ···
⎞ · · · · · · 0 −2 ⎟ .. .. .. . . . 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. .. .. ⎟ . . . 0 ⎟ ⎟ · · · 0 −2 −2 ⎟ ⎟ ··· ··· ··· 0 ⎟ ⎟ ⎟ . .. .. .. .. ⎟ . . . ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ ⎟ .. .. .. . . . 0 ⎠ ··· ··· 0 4
Using Lemma 3 again, yields (m)
τ (Cn ) = ⎛
(m)
τ (Cn ) =
2mn b
2m n−2m b
× det
2a ⎜ ⎜ ( n + 1) ⎜ ⎜ ⎜ 2( n + 2) ⎜ × det⎜ ⎜ ... ⎜ ⎜ ⎜ ⎝ 2( n + 2) ( n + 1)
D F
E 4Im
=
2m n b
( n + 1)
2( n + 2)
2a .. . .. . .. . 2( n + 2)
( n + 3) .. .
Using properties of determinants, we have:
199
.. ..
.
. ···
× det( D − E 4I1m F )
··· 2( n + 2) ( n + 1) .. . ··· 2( n + 2) .. .. .. . . . .. .. . . 2( n + 2) .. .. . . ( n + 1) 2( n + 2) ( n + 1) 2a
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689
⎛
(m)
τ (Cn ) =
2mn b
×
2b m n+3m+2
⎛
=
2m n +1 ( n +3) m m n+3m+2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(2a − n − 1)
0
( n + 3)
0
(2a − n − 1) .. .
0 .. . .. . .. . ···
( n + 3) .. .
..
( n + 3) 0
(2a−n−1) ( n +3)
0
0
(2a−n−1) ( n +3)
. ( n + 3) 1
..
.
..
.
1
..
.
0 .. . .. . .. .
0
1
···
1 .. .
.
..
··· ( n + 3) 0 .. . ··· ( n + 3) .. .. .. . . . .. .. . . ( n + 3) .. .. . . 0 ( n + 3) 0 (2a − n − 1)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎞
··· .. . .. . .. . .. .
1 .. .
1
0
0
..
.
1 .. .
..
.
1
..
.
0
(2a−n−1) ( n +3)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 2, yields: (m)
2( 2an−+n3−1 +m−3) ( n +3) m × [ Tm ( 2a−n−1 −3 mn+3m+2 × n +3 4 2mn+1 × (n + 3)m × [ Tm ( nn+ ) +3 − 1] .
τ (Cn ) = 2m n+1 ×
=
2a−n−1 −1 n +3
2
) − 1]
Using Equation (11), yields the result. (m)
Deﬁnition 4. The pyramid graph Dn is the graph created from the gear graph Gm+1 with vertices {u0 ; u1 , u2 , . . . , um ; w1 , w2 , . . . , wm } with double internal and external edges and m sets of vertices, say, 3 1 1 4 3 4 3 4 j v1 , v2 , . . . , v1n , v21 , v22 , . . . , v2n , . . . , v1m , v2m , . . . , vm n , such that for all i = 1, 2, . . . , n the vertex vi is m adjacent to u j and u j+1 , where j = 1, 2, . . . , m − 1, and vi is adjacent to u1 and um . See Figure 4. X
Z
Y
Y
Y
Y Y
Y Q
Y
Z
Y Q X
Y Q Y Y
Y X
X
Z
Figure 4. The pyramid graph
200
(3) Dn .
Symmetry 2018, 10, 689
√ √ m m (m) Theorem 4. For n ≥ 0, m ≥ 3, τ ( Dn ) = 2mn [(n + 5 + 2 n + 4 ) + (n + 5 − 2 n + 4 ) − m 2( n + 3) ]. Proof. Applying Lemma 1, we have: (m)
τ ( Dn ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ det ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 (mn+2m+1)2
−1 · · · · · · · · · −1 −1 (2m + 1) −1 1 ··· 1 0 (2n + 7) 0 .. .. .. .. .. . . . . 1 . 0 .. .. .. .. .. .. . . . . . 1 . .. .. .. .. .. .. . . . . . . 1 .. .. .. .. .. . . . . . 1 0 −1 0 1 ··· 1 0 (2n + 7) 1 0 0 1 ··· ··· 1 .. .. .. .. .. .. . . . . . 1 . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 1 .. .. .. .. . . . . 1 1 0 0 1 0 1 ··· ··· 1 .. 0 . 0 0 1 ··· 1 .. .. .. .. ... ... ... . . . . .. .. .. . 0 0 1 . 1 . .. .. .. . 1 0 0 . 1 . . . . .. .. .. .. .. .. .. . . . . .. .. .. . 1 0 0 . 1 . .. .. .. .. . 1 1 0 . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. . 1 1 0 . 1 . .. .. .. .. .. .. . . . 1 . . . .. .. .. .. .. .. . . 1 . . . . .. .. .. . 1 1 1 . 0 . .. .. .. .. .. .. ... . . . . . . .. .. .. .. . 1 1 . . 0 . .. .. .. . 0 1 1 . 0 . .. .. .. .. .. .. .. . . . . . . . ··· 1 0 1 1 ··· 0
× det((mn + 2m + 1) I − D c + Ac ) =
··· ··· 1 ··· .. .. . . 0 .. .. . . 1 .. . . .. . . . ... . . . . . . 1 ··· ··· 3 1 ··· .. .. . . 1 . .. .. . . .. . .. .. . . .. . .. .. . . .. 1 ··· ··· 1 0
··· ··· ··· 1 .. .. . . .. .. . . .. .. . . .. .. . . 1 0 ··· ··· .. .. . . .. .. . . .. .. . . .. .. . . ··· 1
1 × (mn+2m+1)2
1 1 0 0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 0 1 ··· 1 ··· ··· 1 ··· 1 1 ··· 1 0 ···
1 0 .. . 1 .. . 1 . 1 .. 0 1 1 1 .. . 1 . . .. .. . . .. .. .. . 1 3 1
···
0
0
···
0
1
···
1
···
1
0
···
0
0
···
0
···
1
1
···
1
0
···
0
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 1
1 .. .
1 .. .
··· ··· .. .. . .
1 .. .
1 .. .
1
1
··· ···
1
1
1 .. .
1 .. .
··· ··· .. .. . .
1 .. .
1 .. .
1 .. . .. . .. . .. . .. . .. . .. .
1 .. . .. . .. . .. . .. . .. . .. .
··· ··· .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . . .. .. . .
1 .. . .. . .. . .. . .. . .. . .. .
1 .. . .. . .. . .. . .. . .. . .. .
1
1
··· ···
1
1
1 .. . 1
1 .. . 1
··· ··· .. .. . . ··· ···
1 .. . 1
1 .. . 1
3 1 .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . 1
··· ···
1
···
1
···
1
···
1
··· ···
1
···
0
···
0
···
1
···
1
1
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 1 1 ··· 1 ··· ··· 1 ··· 1 0 ··· 0 0 ··· ··· 1 1 ··· 1 ··· ··· ··· ··· ··· 1 ··· 1 1 ··· ···
1 .. . .. .
···
···
1 .. . .. .
··· ···
1 1
1 1
···
··· ··· ··· ··· ···
···
1 .. . .. .
··· ···
1 1
···
···
··· ··· ··· ··· ···
1 .. . .. .
··· ··· ··· ··· ··· ··· ··· ··· ··· ···
1 1
··· ··· ··· ··· ···
1 .. . .. .
···
···
1 .. . .. .
··· ···
1 1
1 1
··· ···
···
··· ···
1 .. .
..
.
⎞ 1 ⎟ 0 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ ··· ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ . ⎟ .. ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ ⎟ 1 ⎟ ⎟ .. ⎟ . ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ ... ⎟ ⎟ .. ⎟ ⎟ . ⎟ .. ⎟ ⎟ . ⎟ ⎟ ⎟ 1 ⎠ 3
Let j = (1 · · · 1) be the 1 × n matrix with all one, and Jn the n × n matrix with all one. Set a = 2n + 7 and b = mn + 2m + 1. Then we have: ⎛
(m) τ Dn =
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
2m + 1 −1 .. . .. . .. . .. . −1 1 .. . .. . .. . .. . 1 jt .. . .. . .. . .. . jt
−1 · · · · · · a 0 1 .. .. . . 0 .. .. . . 1 .. .. .. . . . .. .. . . 1 0 1 ··· 0 0 1 .. . 1 0 .. .. .. . . . .. .. .. . . . .. .. . . 1 0 1 ··· 0 0 jt .. t . j 0 .. .. .. . . . .. .. ... . . .. .. . . jt t 0 j ···
· · · · · · −1 ··· 1 0 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. . . 0 1 0 a ··· ··· 1 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. . . 0 ··· 1 0 · · · · · · jt .. .. .. . . . .. .. .. . . . .. .. . . jt .. . ···
..
. jt
0 0
1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
201
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . .. ..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
··· j .. . 0 .. . j .. . . . . ... . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ . .. .. . . .. ⎟ ⎟ . ⎟ .. .. . . . . ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
1 b2
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛
b b .. . .. . .. . .. . b b .. . .. . .. . .. . b bjt .. . .. . .. . .. . bjt
1 ⎜ 1 ⎜ ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ ⎜ . ⎜ .. ⎜ ⎜ . ⎜ ⎜ .. ⎜ . ⎜ ⎜ 1 ⎜ ⎜ ⎜ 1 ⎜ .. ⎜ ⎜ . ⎜ ⎜. . .. 1 ⎜ . = b det⎜ . ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎜ 1 ⎜ ⎜ 1jt ⎜ ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ . ⎜ ⎜ .. ⎜ ⎝ . 1jt ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 = b det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0 0 .. . .. . .. . .. . 0
−1 · · · · · · a 0 1 .. .. . . 0 .. .. . . 1 .. .. ... . . .. .. . . 1 0 1 ··· 0 0 1 .. . 1 0 .. .. .. . . . .. .. .. . . . .. .. . . 1 0 1 ··· 0 0 jt .. . jt 0 .. .. .. . . . .. .. .. . . . .. .. t . . j 0 jt · · ·
· · · · · · −1 ··· 1 0 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. . . 0 1 0 a ··· ··· 1 .. .. . . 1 .. .. .. . . . .. .. . . 1 .. .. . . 0 ··· 1 0 · · · · · · jt .. .. .. . . . . .. .. . . . . .. .. . . jt .. . ···
..
. jt
0 0
−1 · · · · · · · · · · · · −1 0 a 0 1 ··· 1 .. .. .. .. . . . . 0 1 .. .. .. .. .. . . . . 1 . .. .. .. .. ... . . . . 1 .. .. .. .. . . . . 1 0 0 1 ··· 1 0 a 0 0 1 ··· ··· 1 .. .. .. . . . 1 0 1 .. .. .. .. .. .. . . . . . . .. .. .. .. .. . . . . . 1 .. .. .. .. . . . . 1 0 0 0 1 ··· ··· 1 t 0 0 j · · · · · · jt .. . . . .. .. .. jt 0 . .. .. .. .. .. .. . . . . . . .. .. .. .. .. . . . . . jt .. .. .. .. t . . . . j 0 0 jt · · · · · · jt 0
1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt 1 0 0 1 .. . .. . 1 3 1 .. . .. . .. . 1 jt .. . .. . .. . .. . jt
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
.. .. .. .. . . . . ··· ··· ··· ···
··· ··· ··· ··· 1 ··· ··· 1 .. .. .. . . . 0 .. .. .. . . . 0 .. .. .. .. . . . . .. .. . . 0 0 ··· ··· 1 0 1 ··· ··· ··· .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 1 ··· ··· ··· ··· .. .. .. .. . . . . ..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.. .. .. .. . . . . ··· ··· ··· ···
−1 ··· ··· ··· ··· −1 1 ··· 2 ··· 2 1 −1 0 ( a + 1) 1 .. .. .. .. . . . . 1 2 −1 −1 .. .. .. .. .. .. . . . . . 2 . 0 .. .. .. .. .. .. ... . . . . . . 2 .. .. .. .. .. .. . . . . . 2 . 1 1 2 ··· 2 1 ( a + 1) 0 · · · 2 2 0 1 1 2 ··· ··· .. .. .. .. . . . . 2 1 2 0 .. .. .. .. .. .. .. .. . . . . . . . . .. .. .. .. .. .. .. . . . . . . . 2 .. .. .. .. .. .. . . . . . . 2 1 1 2 ··· ··· 2 1 0 ··· jt 2jt · · · · · · 2jt 0 ··· jt .. . .. . . . .. .. .. .. . . 2jt jt .. .. .. .. .. .. .. .. . . . . . . . . .. .. .. .. .. .. .. . . . . . . 2jt . .. .. .. .. .. .. t t . . . . . 2j . j jt 2jt · · · · · · 2jt jt 0 ···
202
1 0 1 .. . .. . 1 0 1 .. . .. . .. . 1 3 jt .. . .. . .. . .. . jt
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . ... . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· j .. . 0 .. . j .. . . . . .. . . . . j ··· j ··· .. . . . . .. . . . . .. . . . . .. . . . . j ··· j 0
··· ··· .. . .. . .. . .. . ··· ··· .. . ..
.
..
.
..
. ···
2Im n + Jm n
··· ··· ··· 1 j ··· ··· · · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . 0 −j .. .. .. .. .. .. . . . . . . 0 .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. . . −1 0 . . . · · · 0 −1 −1 0 · · · · · · ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 0 ··· ··· 0 2 0 ··· ··· ··· ··· ··· 0 .. .. .. .. . . . . .. .. .. .. . . . . 2Im n .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 0
⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . j ⎟ . ⎟ ⎟ . .. .. ⎟ . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ··· ··· j ··· j 0 ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ ⎟ . .. .. ⎟ . . .. ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . j ⎟ ⎟ j 0 0 ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ . ⎟ .. .. . . .. ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ . .. .. . . .. ⎟ ⎟ ⎟ ··· ··· j ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ··· ··· j · · · 0 −j ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 = b det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
( a + 1) 1 2 .. . 2 1 1 2 .. . .. . 2 1 jt 2jt .. . .. . 2jt jt
1 .. . .. . .. . .. . 2 1 1 .. . .. . .. . 2 jt jt .. . ..
··· 2 1 −1 0 .. .. . . 2 −1 −1 .. .. .. .. . . . . 0 .. .. .. .. . . . . 2 .. .. .. .. . . . . 1 2 1 ( a + 1) 0 · · · ··· ··· 2 2 0 .. .. .. . . . 2 0 .. .. .. .. ... . . . . .. .. .. .. . . . . 2 . .. .. .. . . . . . 1 ··· 2 1 0 ··· ··· ··· 2jt 0 ··· .. .. .. .. .. . . . . . .. .. .. .. .. . . . . . .. .. .. .. . . . 2jt . .. .. .. .. . . . . jt · · · 2jt jt 0 ···
2 .. . .. . .. . .. . ··· 2 .. . .. . .. . .. . ··· 2jt .. . .. . .. . .. . ···
.
..
. 2jt
· · · · · · 0 −1 − j 0 ··· .. .. .. .. .. . . . . . 0 −j .. .. .. .. .. .. . . . . . . 0 .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. . . −1 0 . . . · · · 0 −1 −1 0 · · · · · · ··· ··· ··· 0 0 ··· ··· .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. .. . . . . . . . .. .. .. .. .. .. . . . . . . 0 ··· ··· 0 2 0 ··· ··· ··· ··· ··· 0 .. .. .. .. . . . . .. .. .. .. . . . . 2Im n .. .. .. .. . . . . .. .. .. .. . . . . ··· ··· ··· 0
⎞ · · · 0 −j ⎟ .. .. . 0 ⎟ . ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. ⎟ .. .. . . . ⎟ ⎟ ⎟ .. .. . . 0 ⎟ ⎟ 0 −j −j ⎟ ⎟ ⎟ ··· ··· 0 ⎟ .. ⎟ .. .. ⎟ . . . ⎟ ⎟ .. .. ... ⎟ ⎟ . . ⎟ ⎟ . .. .. .. ⎟ . . ⎟ ⎟ . ⎟ .. .. . . . . ⎟ ⎟ ··· ··· 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 3, yields (m) τ ( Dn ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 mn − 2m ⎜ = b2 × 2 × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
=
1 b
× det
A C
B 2Imn
=
1 b
× det( A − B 2I1mn C ) × 2mn
(2a + 2n + 2)
3n + 2
4( n + 1)
···
4( n + 1)
3n + 2
−2
3n + 2
(2a + 2n + 2) 3n + 4 .. . .. . 4( n + 1) 2 .. . .. . .. . .. . 4
4( n + 1) .. . .. . .. . 4( n + 1) ··· .. . .. . .. . .. . ···
··· .. . .. . .. . 3n + 2 ··· .. .
4( n + 1) .. .
−2
4( n + 1) .. .
3n + 2 .. . .. . .. . ··· 4 .. . .. . .. . .. . ···
4( n + 1) 3n + 2 2 4 .. . .. . 4 2
4( n + 1)
..
.
3n + 2 (2a + 2n + 2) 4 .. . .. .
..
.
4
. 4
2 2
..
0 .. . .. . 0 4 0 .. . .. . .. . 0
0 .. . .. . .. . .. . ··· 0 .. . .. . .. . .. . ···
⎞ · · · · · · 0 −2 ⎟ .. .. .. . . . 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. .. .. ⎟ . . . 0 ⎟ ⎟ · · · 0 −2 −2 ⎟ ⎟ ··· ··· ··· 0 ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ .. ⎟ .. .. .. ⎟ . . . . ⎟ ⎟ .. ⎟ .. .. .. . . . . ⎟ ⎟ ⎟ .. .. .. . . . 0 ⎠ ··· ··· 0 4
Using Lemma 3, yields (m)
τ ( Dn ) = ⎛
(m)
τ ( Dn ) =
2mn b
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎝
2m n−2m b
× det
(2a + 2n + 4) (3n + 5) 4( n + 2) .. . 4( n + 2) (3n + 5)
A C
B 4Im
=
(3n + 5)
2m n b
× det( A − B 4I1m C )
4( n + 2)
··· 4( n + 2) (3n + 5) .. . (2a + 2n + 4) (3n + 5) ··· 4( n + 2) .. .. .. .. .. . . . . . .. .. .. .. . . . . 4( n + 2) .. .. .. .. . . . . (3n + 5) 4( n + 2) ··· 4(n + 2) (3n + 5) (2a + 2n + 4)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Straightforward inducement using properties of determinants, we get: ⎛
(m)
τ ( Dn ) =
2mn b
×
4b m n+3m+4
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(2a − n − 1)
0
( n + 3)
0
(2a − n − 1) .. .
0 .. . .. . .. . ···
( n + 3) .. . ( n + 3) 0
.. ..
.
. ( n + 3)
203
··· ( n + 3) 0 .. . ··· ( n + 3) .. .. .. . . . .. .. . . ( n + 3) .. .. . . 0 ( n + 3) 0 (2a − n − 1)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Symmetry 2018, 10, 689 ⎛
=
2m n +2 ( n + 3) m m n+3m+4
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ × det⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(2a−n−1) ( n +3)
0
0
(2a−n−1) ( n +3)
1
..
.
..
.
1
..
.
0 .. . .. . .. .
0
1
···
1 .. .
⎞
··· .. . .. . .. . .. .
1 .. .
1
0
0
..
.
1 .. .
..
.
1
..
.
0
(2a−n−1) ( n +3)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Using Lemma 2, yields: (m)
τ ( Dn ) = 2 m n + 2 ×
( n +3) m mn+3m+4
×
2( 2an−+n3−1 +m−3) 2a−n−1 −3 n +3
× [ Tm (
2a−n−1 −1 n +3
2
5 ) − 1] = 2mn+1 × (n + 3)m × [ Tm ( nn+ +3 ) − 1].
Using Equation (11), yields the result. 4. Numerical Results The following Table 1 illustrates some values of the number of spanning trees of studied pyramid graphs. Table 1. Some values of the number of spanning trees of studied pyramid graphs. m
n
3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
(m)
τ (Pn
)
50 1024 15, 488 200, 704 2, 367, 488 26, 214, 400 192 11, 520 458, 752 14, 745, 600 415, 236, 096 10, 687, 086, 592 722 123, 904 12, 781, 568 1, 007, 681, 536 67, 194, 847, 232 3, 995, 393, 327, 104
(m)
τ ( An
)
196 3200 43, 264 524, 288 5, 914, 624 63, 438, 848 1152 49, 152 1, 638, 400 47, 185, 920 1, 233, 125, 376 30, 064, 771, 072 6724 739, 328 59, 969, 536 4, 060, 086, 272 243, 609, 370, 624 243, 609, 370, 624
(m)
τ ( Bn
)
242 3136 36, 992 409, 600 4, 333, 568 44, 302, 336 1792 57, 600 1, 622, 016 41, 746, 432 1, 006, 632, 960 23, 102, 226, 432 12, 482 984, 064 65, 619, 968 3, 901, 751, 296 213, 408, 284, 672 10, 953, 240, 346, 624
(m)
τ (Cn
)
676 8192 92, 416 991, 232 10, 240, 000 102, 760, 448 6400 184, 320 4, 816, 896 117, 440, 512 2, 717, 908, 992 60, 397, 977, 600 58, 564 3, 964, 928 237, 899, 776 13, 088, 325, 632 674, 448, 277, 504 33, 019, 708, 571, 648
5. Conclusions The number of spanning trees τ ( G ) in graphs (networks) is an important invariant. The computation of this number is not only beneﬁcial from a mathematical (computational) standpoint, but it is also an important measure of the reliability of a network and electrical circuit layout. Some computationally laborious problems such as the traveling salesman problem can be resolved approximately by using spanning trees. In this paper, we deﬁne some classes of pyramid graphs created by a gear graph and we have studied the problem of computing the number of spanning trees of these graphs. Author Contributions: All authors contributed equally to this work. Funding Acquisition, J.B.L.; Methodology, J.B.L. and S.N.D. Daoud; Writing—Original Draft, J.B.L. and S.N.D. Daoud; All authors read and approved the ﬁnal manuscript.
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Funding: The work was partially supported by the China Postdoctoral Science Foundation under grant No. 2017M621579 and the Postdoctoral Science Foundation of Jiangsu Province under grant No. 1701081B, Project of Anhui Jianzhu University under Grant no. 2016QD116 and 2017dc03, Anhui Province Key Laboratory of Intelligent Building & Building Energy Saving. Acknowledgments: The authors are grateful to the anonymous reviewers for their helpful comments and suggestions for improving the original version of the paper. Conﬂicts of Interest: The authors declare that there are no conﬂicts of interest regarding the publication of this paper.
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Applegate, D.L.; Bixby, R.E.; Chvátal, V.; Cook, W.J. The Traveling Salesman Problem: A Computational Study; Princeton University Press: Princeton, NJ, USA, 2006. Cvetkovi˘e, D.; Doob, M.; Sachs, H. Spectra of Graphs: Theory and Applications, 3rd ed.; Johann Ambrosius Barth: Heidelberg, Germany, 1995. Kirby, E.C.; Klein, D.J.; Mallion, R.B.; Pollak, P.; Sachs, H. A theorem for counting spanning trees in general chemical graphs and its particular application to toroidal fullerenes. Croat. Chem. Acta 2004, 77, 263–278. Boesch, F.T.; Satyanarayana, A.; Suffel, C.L. A survey of some network reliability analysis and synthesis results. Networks 2009, 54, 99–107. [CrossRef] Boesch, F.T. On unreliability polynomials and graph connectivity in reliable network synthesis. J. Graph Theory 1986, 10, 339–352. [CrossRef] Wu, F.Y. Number of spanning trees on a Lattice. J. Phys. A 1977, 10, 113–115. [CrossRef] Zhang, F.; Yong, X. Asymptotic enumeration theorems for the number of spanning trees and Eulerian trail in circulant digraphs & graphs. Sci. China Ser. A 1999, 43, 264–271. Chen, G.; Wu, B.; Zhang, Z. Properties and applications of Laplacian spectra for Koch networks. J. Phys. A Math. Theor. 2012, 45, 025102. Atajan, T.; Inaba, H. Network reliability analysis by counting the number of spanning trees. In Proceedings of the IEEE International Symposium on Communications and Information Technology, ISCIT 2004, Sapporo, Japan, 26–29 October 2004; pp. 601–604. Brown, T.J.N.; Mallion, R.B.; Pollak, P.; Roth, A. Some methods for counting the spanning trees in labelled molecular graphs, examined in relation to certain fullerenes. Discret. Appl. Math. 1996, 67, 51–66. [CrossRef] Kirchhoff, G.G. Uber die Auﬂosung der Gleichungen, auf welche man be ider Untersuchung der Linearen Verteilung galvanischer Storme gefuhrt wird. Ann. Phys. Chem. 1847, 72, 497–508. [CrossRef] Kelmans, A.K.; Chelnokov, V.M. A certain polynomials of a graph and graphs with an extermal number of trees. J. Comb. Theory B 1974, 16, 197–214. [CrossRef] Biggs, N.L. Algebraic Graph Theory, 2nd ed.; Cambridge University Press: Cambridge, UK, 1993; p. 205. Daoud, S.N. The deletioncontraction method for counting the number of spanning trees of graphs. Eur. Phys. J. Plus 2015, 130, 217. [CrossRef] Shang, Y. On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdividedline graphs. Open Math. 2016, 14, 641–648. [CrossRef] Bozkurt, S¸ .B.; Bozkurt, D. On the Number of Spanning Trees of Graphs. Sci. World J. 2014, 2014, 294038. [CrossRef] [PubMed] Daoud, S.N. Number of Spanning Trees in Different Product of Complete and Complete Tripartite Graphs. ARS Comb. 2018, 139, 85–103. Daoud, S.N. Complexity of Graphs Generated by Wheel Graph and Their Asymptotic Limits. J. Egypt. Math. Soc. 2017, 25, 424–433. [CrossRef] Daoud, S.N. Chebyshev polynomials and spanning tree formulas. Int. J. Math. Comb. 2012, 4, 68–79. Zhang, Y.; Yong, X.; Golin, M.J. Chebyshev polynomials and spanning trees formulas for circulant and related graphs. Discret. Math. 2005, 298, 334–364. [CrossRef]
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21. 22.
Daoud, S.N. On a class of some pyramid graphs and Chebyshev polynomials. J. Math. Probl. Eng. Hindawi Publ. Corp. 2013, 2013, 820549. Marcus, M. A Survey of Matrix Theory and Matrix Inequalities; University Allyn and Bacon. Inc.: Boston, MA, USA, 1964. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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SS symmetry Article
Novel ThreeWay Decisions Models with MultiGranulation Rough Intuitionistic Fuzzy Sets ZhanAo Xue 1,2, *, DanJie Han 1,2, *, MinJie Lv 1,2 and Min Zhang 1,2 1 2
*
College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China; [email protected] (M.J.L.); [email protected] (M.Z.) Engineering Lab of Henan Province for Intelligence Business & Internet of Things, Henan Normal University, Xinxiang 453007, China Correspondence: [email protected] (Z.A.X.); [email protected] (D.J.H.)
Received: 27 October 2018; Accepted: 16 November 2018; Published: 21 November 2018
Abstract: The existing construction methods of granularity importance degree only consider the direct inﬂuence of single granularity on decisionmaking; however, they ignore the joint impact from other granularities when carrying out granularity selection. In this regard, we have the following improvements. First of all, we deﬁne a more reasonable granularity importance degree calculating method among multiple granularities to deal with the above problem and give a granularity reduction algorithm based on this method. Besides, this paper combines the reduction sets of optimistic and pessimistic multigranulation rough sets with intuitionistic fuzzy sets, respectively, and their related properties are shown synchronously. Based on this, to further reduce the redundant objects in each granularity of reduction sets, four novel kinds of threeway decisions models with multigranulation rough intuitionistic fuzzy sets are developed. Moreover, a series of concrete examples can demonstrate that these joint models not only can remove the redundant objects inside each granularity of the reduction sets, but also can generate much suitable granularity selection results using the designed comprehensive score function and comprehensive accuracy function of granularities. Keywords: threeway decisions; intuitionistic fuzzy sets; multigranulation rough intuitionistic fuzzy sets; granularity importance degree
1. Introduction Pawlak [1,2] proposed rough sets theory in 1982 as a method of dealing with inaccuracy and uncertainty, and it has been developed into a variety of theories [3–6]. For example, the multigranulation rough sets (MRS) model is one of the important developments [7,8]. The MRS can also be regarded as a mathematical framework to handle granular computing, which is proposed by Qian et al. [9]. Thereinto, the problem of granularity reduction is a vital research aspect of MRS. Considering the test cost problem of granularity structure selection in data mining and machine learning, Yang et al. constructed two reduction algorithms of costsensitive multigranulation decisionmaking system based on the deﬁnition of approximate quality [10]. Through introducing the concept of distribution reduction [11] and taking the quality of approximate distribution as the measure in the multigranulation decision rough sets model, Sang et al. proposed an αlower approximate distribution reduction algorithm based on multigranulation decision rough sets, however, the interactions among multiple granularities were not considered [12]. In order to overcome the problem of updating reduction, when the largescale data vary dynamically, Jing et al. developed an incremental attribute reduction approach based on knowledge granularity with a multigranulation view [13]. Then other multigranulation reduction methods have been put forward one after another [14–17].
Symmetry 2018, 10, 662; doi:10.3390/sym10110662
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Symmetry 2018, 10, 662
The notion of intuitionistic fuzzy sets (IFS), proposed by Atanassov [18,19], was initially developed in the framework of fuzzy sets [20,21]. Within the previous literature, how to get reasonable membership and nonmembership functions is a key issue. In the interest of dealing with fuzzy information better, many experts and scholars have expanded the IFS model. Huang et al. combined IFS with MRS to obtain intuitionistic fuzzy MRS [22]. On the basis of fuzzy rough sets, Liu et al. constructed coveringbased multigranulation fuzzy rough sets [23]. Moreover, multigranulation rough intuitionistic fuzzy cut sets model was structured by Xue et al. [24]. In order to reduce the classiﬁcation errors and the limitation of ordering by single theory, they further combined IFS with graded rough sets theory based on dominance relation and extended them to a multigranulation perspective. [25]. Under the optimistic multigranulation intuitionistic fuzzy rough sets, Wang et al. proposed a novel method to solve multiple criteria group decisionmaking problems [26]. However, the above studies rarely deal with the optimal granularity selection problem in intuitionistic fuzzy environments. The measure of similarity between intuitionistic fuzzy sets is also one of the hot areas of research for experts, and some similarity measures about IFS are summarized in references [27–29], whereas these metric formulas cannot measure the importance degree of multiple granularities in the same IFS. For further explaining the semantics of decisiontheoretic rough sets (DTRS), Yao proposed a threeway decisions theory [30,31], which vastly pushed the development of rough sets. As a risk decisionmaking method, the key strategy of threeway decisions is to divide the domain into acceptance, rejection, and noncommitment. Up to now, researchers have accumulated a vast literature on its theory and application. For instance, in order to narrow the applications limits of threeway decisions model in uncertainty environment, Zhai et al. extended the threeway decisions models to tolerance rough fuzzy sets and rough fuzzy sets, respectively, the target concepts are relatively extended to tolerance rough fuzzy sets and rough fuzzy sets [32,33]. To accommodate the situation where the objects or attributes in a multiscale decision table are sequentially updated, Hao et al. used sequential threeway decisions to investigate the optimal scale selection problem [34]. Subsequently, Luo et al. applied threeway decisions theory to incomplete multiscale information systems [35]. With respect to multiple attribute decisionmaking, Zhang et al. study the inclusion relations of neutrosophic sets in their case in reference [36]. For improving the classiﬁcation correct rate of threeway decisions, Zhang et al. proposed a novel threeway decisions model with DTRS by considering the new risk measurement functions through the utility theory [37]. Yang et al. combined threeway decisions theory with IFS to obtain novel threeway decision rules [38]. At the same time, Liu et al. explored the intuitionistic fuzzy threeway decision theory based on intuitionistic fuzzy decision systems [39]. Nevertheless, Yang et al. [38] and Liu et al. [39] only considered the case of a single granularity, and did not analyze the decisionmaking situation of multiple granularities in an intuitionistic fuzzy environment. The DTRS and threeway decisions theory are both used to deal with decisionmaking problems, so it is also enlightening for us to study threeway decisions theory through DTRS. An extension version that can be used to multiperiods scenarios has been introduced by Liang et al. using intuitionistic fuzzy decision theoretic rough sets [40]. Furthermore, they introduced the intuitionistic fuzzy point operator into DTRS [41]. The threeway decisions are also applied in multiple attribute group decision making [42], supplier selection problem [43], clustering analysis [44], cognitive computer [45], and so on. However, they have not applied the threeway decisions theory to the optimal granularity selection problem. To solve this problem, we have expanded the threeway decisions models. The main contributions of this paper include four points: (1) The new granularity importance degree calculating methods among multiple granularities ,Δ (i.e., sig,Δ in ( Ai , A , D ) and sig out ( Ai , A , D )) are given respectively, which can generate more discriminative granularities. (2) Optimistic optimistic multigranulation rough intuitionistic fuzzy sets (OOMRIFS) model, optimistic pessimistic multigranulation rough intuitionistic fuzzy sets (OIMRIFS) model,
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Symmetry 2018, 10, 662
pessimistic optimistic multigranulation rough intuitionistic fuzzy sets (IOMRIFS) model and pessimistic pessimistic multigranulation rough intuitionistic fuzzy sets (IIMRIFS) model are constructed by combining intuitionistic fuzzy sets with the reduction of the optimistic and pessimistic multigranulation rough sets. These four models can reduce the subjective errors caused by a single intuitionistic fuzzy set. (3) We put forward four kinds of threeway decisions models based on the proposed four multigranulation rough intuitionistic fuzzy sets (MRIFS), which can further reduce the redundant objects in each granularity of reduction sets. (4) Comprehensive score function and comprehensive accuracy function based on MRIFS are constructed. Based on this, we can obtain the optimal granularity selection results. The rest of this paper is organized as follows. In Section 2, some basic concepts of MRS, IFS, and threeway decisions are brieﬂy reviewed. In Section 3, we propose two new granularity importance degree calculating methods and a granularity reduction Algorithm 1. At the same time, a comparative example is given. Four novel MRIFS models are constructed in Section 4, and the properties of the four models are veriﬁed by Example 2. Section 5 proposes some novel threeway decisions models based on above four new MRIFS, and the comprehensive score function and comprehensive accuracy function based on MRIFS are built. At the same time, through Algorithm 2, we make the optimal granularity selection. In Section 6, we use Example 3 to study and illustrate the threeway decisions models based on new MRIFS. Section 7 concludes this paper. 2. Preliminaries The basic notions of MRS, IFS, and threeway decisions theory are brieﬂy reviewed in this section. Throughout the paper, we denote U as a nonempty object set, i.e., the universe of discourse and A = { A1 , A2 , · · · , Am } is an attribute set. Suppose IS =< U, A, V, f > is a consistent information system, Deﬁnition 1 ([9]). A = { A1 , A2 , · · · , Am } is an attribute set. And R Ai is an equivalence relation generated by A. [ x ] Ai is the equivalence class of R Ai , ∀ X ⊆ U, the lower and upper approximations of optimistic multigranulation rough sets (OMRS) of X are deﬁned by the following two formulas: O
m
∑ Ai ( X ) = { x ∈ U [ x ] A1 ⊆ X ∨ [ x ] A2 ⊆ X ∨ [ x ] A3 ⊆ X . . . ∨ [ x ] Am ⊆ X };
i =1
O
m
O
m
∑ Ai ( X ) = ∼ ( ∑ Ai ( ∼ X )).
i =1
i =1
m
O
m
O
where ∨ is a disjunction operation, ∼ X is a complement of X, if ∑ Ai ( X ) = ∑ Ai ( X ), the pair i =1
m
O
m
i =1
O
( ∑ Ai ( X ), ∑ Ai ( X )) is referred to as an optimistic multigranulation rough set of X. i =1
i =1
Deﬁnition 2 ([9]). Let IS =< U, A, V, f > be an information system, where A = { A1 , A2 , · · · , Am } is an attribute set, and R Ai is an equivalence relation generated by A. [ x ] Ai is the equivalence class of R Ai , ∀ X ⊆ U, the pessimistic multigranulation rough sets (IMRS) of X with respect to A are deﬁned as follows: m
I
∑ Ai ( X ) = { x ∈ U [ x ] A1 ⊆ X ∧ [ x ] A2 ⊆ X ∧ [ x ] A3 ⊆ X ∧ . . . ∧ [ x ] Am ⊆ X };
i =1
m
I
m
I
∑ Ai ( X ) = ∼ ( ∑ Ai ( ∼ X )).
i =1
i =1
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Symmetry 2018, 10, 662
m
I
where [ x ] Ai (1 ≤ i ≤ m) is equivalence class of x for Ai , ∧ is a conjunction operation, if ∑ Ai ( X ) = i =1
m
I
m
I
I
m
∑ Ai ( X ), the pair ( ∑ Ai ( X ), ∑ Ai ( X )) is referred to as a pessimistic multigranulation rough set of X.
i =1
i =1
i =1
Deﬁnition 3 ([18,19]). Let U be a ﬁnite nonempty universe set, then the IFS E in U are denoted by: E = {< x, μ E ( x ), νE ( x ) >  x ∈ U }, where μ E ( x ) : U → [0, 1] and νE ( x ) : U → [0, 1] . μ E ( x ) and νE ( x ) are called membership and nonmembership functions of the element x in E with 0 ≤ μ E ( x ) + νE ( x ) ≤ 1. For ∀ x ∈ U, the hesitancy degree function is deﬁned as π E ( x ) = 1 − μ E ( x ) − νE ( x ), obviously, π E ( x ) : U → [0, 1] . Suppose ∀ E1 , E2 ∈ IFS(U ), the basic operations of E1 and E2 are given as follows: (1) (2) (3) (4) (5)
E1 ⊆ E2 ⇔ μ E1 ( x ) ≤ μ E2 ( x ), νE1 ( x ) ≥ νE2 ( x ), ∀ x ∈ U; A = B ⇔ μ A ( x ) = μ B ( x ), νA ( x ) = νB ( x ), ∀ x ∈ U; E1 ∪ E2 = {< x, max{μ E1 ( x ), μ E2 ( x )}, min{νE1 ( x ), νE2 ( x )} >  x ∈ U }; (4) E1 ∩ E2 = {< x, min{μ E1 ( x ), μ E2 ( x )}, max{νE1 ( x ), νE2 ( x )} >  x ∈ U }; (5) ∼ E1 = {< x, νE1 ( x ), μ E1 ( x ) >  x ∈ U }.
Deﬁnition 4 ([30,31]). Let U = { x1 , x2 , · · · , xn } be a universe of discourse, ξ = {ω P , ω N , ω B } represents the decisions of dividing an object x into receptive POS( X ), rejective NEG ( X ), and boundary regions BND ( X ), respectively. The cost functions λ PP , λ NP and λ BP are used to represent the three decision making costs of ∀ x ∈ U, and the cost functions λ PN , λ NN and λ BN are used to represent the three decisionmaking costs of ∀x ∈ / U, as shown in Table 1. Table 1. Cost matrix of decision actions. Decision Functions
Decision Actions ωP ωB ωN
X
∼X
λ PP λ BP λ NP
λ PN λ BN λ NN
According to the minimumrisk principle of Bayesian decision procedure, threeway decisions rules can be obtained as follows: (P): If P( X [ x ]) ≥ α, then x ∈ POS( X ); (N): If P( X [ x ]) ≤ β, then x ∈ NEG ( X ); (B): If β < P( X [ x ]) < α, then x ∈ BND ( X ). Here α, β and γ represent respectively: α=
λ PN − λ BN ; (λ PN − λ BN ) + (λ BP − λ PP )
β=
λ BN − λ NN ; (λ BN − λ NN ) + (λ NP − λ BP )
γ=
λ PN − λ NN . (λ PN − λ NN ) + (λ NP − λ PP )
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3. Granularity Reduction Algorithm Derives from Granularity Importance Degree Deﬁnition 5 ([10,12]). Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of condition attributes C. U/D = { X1 , X2 , · · · , Xs } is the partition induced by the decision attributes D, then approximation quality of U/D about granularity set A is deﬁned as:
γ( A, D ) =
& ( )& & & Δ m & & &∪ ∑ Ai ( Xt )1 ≤ t ≤ s & & & i =1
U 
.
where  X  denotes the cardinal number of set X. Δ ∈ {O, I } represents two cases of optimistic and pessimistic multigranulation rough sets, the same as the following. Deﬁnition 6 ([12]). Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A, X ∈ U/D, (1) If (2) If
m
∑
i =1,Ai ∈ A m
∑
i =1,Ai ∈ A
Δ
Ai ( X ) =
m
Ai ( X ) =
Ai ( X ), then A is important in A for X;
m
Ai ( X ), then A is not important in A for X.
i =1,Ai
Δ
Δ
∑
∈ A− A
∑
i =1,Ai ∈ A− A
Δ
Deﬁnition 7 ([10,12]). Suppose DIS = (U, C ∪ D, V, f ) is a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A. ∀ Ai ∈ A, on the granularity sets A, the internal importance degree of Ai for D can be deﬁned as follows: Δ ( Ai , A, D ) = γ( A, D ) − γ( A − { Ai }, D ). sigin
Deﬁnition 8 ([10,12]). Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A. ∀ Ai ∈ A − A, on the granularity sets A, the external importance degree of Ai for D can be deﬁned as follows: Δ ( Ai , A, D ) = γ( Ai ∪ A, D ) − γ( A, D ). sigout
Theorem 1. Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A. (1) For ∀ Ai ∈ A, on the basis of attribute subset family A , the granularity importance degree of Ai in A with respect to D is expressed as follows: Δ ( Ai , A , D ) = sigin
1 Δ Δ sigin ({ Ak , Ai }, A, D ) − sigin ( Ak , A − { Ai }, D ). m − 1∑
where 1 ≤ k ≤ m, k = i, the same as the following. (2) For ∀ Ai ∈ A − A, on the basis of attribute subset family A , the granularity importance degree of Ai in A − A with respect to D, we have: Δ ( A i , A , D ) = sigout
1 Δ Δ sigout ({ Ak , Ai }, { Ai } ∪ A, D ) − sigout ( Ak , A, D ). m − 1∑
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Symmetry 2018, 10, 662
Proof. (1) According to Deﬁnition 7, then Δ
sigin ( Ai , A, D )
= γ( A, D ) − γ( A − { Ai }, D ) −1 = m m−1  γ ( A , D ) − γ ( A − { Ai }, D ) + ∑  γ ( A − { Ak , Ai }, D ) − γ ( A − { Ak , Ai }, D ) 1 = m− 1 ∑ ( γ ( A , D ) − γ ( A − { Ak , Ai }, D ) − ( γ ( A − { Ai }, D ) − γ ( A − { Ak , Ai }, D )) Δ Δ 1 = m− 1 ∑  sigin ({ Ak , Ai }, A , D ) − sigin ( Ak , A − { Ai }, D ).
(2) According to Deﬁnition 8, we can get: Δ
sigout ( Ai , A, D )
= γ({ Ai } ∪ A, D ) − γ( A, D ) −1 = m m−1  γ ({ Ai } ∪ A , D ) − γ ( A , D ) − ∑  γ ( A − { Ak }, D ) − γ ( A − { Ak }, D ) 1 = m− 1 ∑ ( γ ({ Ai } ∪ A , D ) − γ ( A − { Ak }, D ) − ( γ ( A − { Ak }, D ) − γ ( A , D )) Δ Δ 1 = m− 1 ∑  sigout ({ Ak , Ai }, { Ai } ∪ A , D ) − sigout ( Ak , A , D ).
In Deﬁnitions 7 and 8, only the direct effect of a single granularity on the whole granularity sets is given, without considering the indirect effect of the remaining granularities on decisionmaking. The following Deﬁnitions 9 and 10 synthetically analyze the interdependence between multiple granularities and present two new methods for calculating granularity importance degree. Deﬁnition 9. Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A. ∀ Ai , Ak ∈ A, on the attribute subset family, A, the new internal importance degree of Ai relative to D is deﬁned as follows: Δ sig,Δ in ( Ai , A , D ) = sigin ( Ai , A , D ) + Δ ( A , A , D ) and sigin i
1 Δ m−1 ∑  sigin ( Ak ,
1 Δ Δ sigin ( Ak , A − { Ai }, D ) − sigin ( Ak , A, D ). m − 1∑
Δ ( A , A , D ) respectively indicate the direct A − { Ai }, D ) − sigin k Δ
Δ
and indirect effects of granularity Ai on decisionmaking. When sigin ( Ak , A − { Ai }, D ) − sigin ( Ak , A, D ) > 0 is satisﬁed, it is shown that the granularity importance degree of Ak is increased by the addition of Ai in attribute subset A − { Ai }, so the granularity importance degree of Ak should be added to Ai . Therefore, 1 Δ Δ when there are m subattributes, we should add m− 1 ∑  sigin ( Ak , A − { Ai }, D ) − sigin ( Ak , A , D ) to the granularity importance degree of Ai . Δ Δ If sigin ( Ak , A − { Ai }, D ) − sigin ( Ak , A, D ) = 0 and k = i, then it shows that there is no interaction Δ
between granularity Ai and other granularities, which means sig,Δ in ( Ai , A , D ) = sigin ( Ai , A , D ).
Deﬁnition 10. Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } be m subattributes of C, A ⊆ A. ∀ Ai ∈ A − A, the new external importance degree of Ai relative to D is deﬁned as follows: Δ sig,Δ out ( Ai , A , D ) = sigout ( Ai , A , D ) +
1 Δ Δ sigout ( Ak , A, D ) − sigout ( Ak , { Ai } ∪ A, D ). m − 1∑
Similarly, the new external importance degree calculation formula has a similar effect. Theorem 2. Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } be m subattributes of C, A ⊆ A, ∀ Ai ∈ A. The improved internal importance can be rewritten as: sig,Δ in ( Ai , A , D ) =
Δ 1 sig ( A , A − { Ak }, D ). m − 1 ∑ in i
212
Symmetry 2018, 10, 662
Proof. sig,Δ in ( Ai , A , D )
Δ
= sigin ( Ai , A, D ) + =
Δ 1 m−1 ∑  sigin ( Ak ,
Δ
A − { Ai }, D ) − sigin ( Ak , A, D )
m −1 1 m−1  γ ( A , D ) − γ ( A − { Ai }, D ) + m−1 ∑  γ ( A − { Ai }, D )−
γ( A − { Ak , Ai }, D ) − γ( A, D ) − γ( A − { Ak }, D )
= =
1 m −1 ∑  γ ( A − { A k }, D ) − γ ( A − { A k , Δ 1 m−1 ∑ sigin ( Ai , A − { Ak }, D ).
Ai }, D )
Theorem 3. Let DIS = (U, C ∪ D, V, f ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A. The improved external importance can be expressed as follows: sig,Δ out ( Ai , A , D ) =
Δ 1 sig ( A , { Ak } ∪ A, D ). m − 1 ∑ out i
Proof. sig,Δ out ( Ai , A , D )
Δ
= sigout ( Ai , A, D ) + =
m −1 m−1  γ ({ Ai } ∪
Δ 1 m−1 ∑ ( sigout ( Ak ,
A, D ) − γ( A, D ) +
Δ
A, D ) − sigout ( Ak , { Ai } ∪ A, D ))
1 m−1 ∑  γ ( A , D ) − γ ({ Ak } ∪
A, D )−
γ({ Ai } ∪ A, D ) = =
1 m−1 ∑  γ ({ Ai , Ak } ∪ A , D ) − γ ({ Ai } ∪ Δ 1 m−1 ∑ sigout ( Ai , { Ak } ∪ A , D ).
A, D )
Δ
Δ ( A , A − { A }, D ) = 0 ( sig Theorems 2 and 3 show that when sigin i k out ( Ai , { Ak } ∪ A , D ) = 0) ,Δ is satisﬁed, having sig,Δ ( A , A , D ) = 0 ( sig ( A , A , D ) = 0 ) . And each granularity importance i i out in degree is calculated on the basis of removing Ak from A, which makes it more convenient for us to choose the required granularity. According to [10,12], we can get optimistic and pessimistic multigranulation lower approximations LO and L I . The granularity reduction algorithm based on improved granularity importance degree is derived from Theorems 2 and 3, as shown in Algorithm 1.
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Symmetry 2018, 10, 662
Algorithm 1. Granularity reduction algorithm derives from granularity importance degree Input: DIS = (U, C ∪ D, V, f ), A = { A1 , A2 , · · · , Am } are m subattributes of C, A ⊆ A, ∀ Ai ∈ A, U/D = { X1 , X2 , · · · , Xs }; Δ Output: A granularity reduction set Ai of this information system. Δ
1: set up Ai ← φ , 1 ≤ h ≤ m; 2: compute U/D, optimistic and pessimistic multigranulation lower approximations LΔ ; 3: for ∀ Ai ∈ A 4: compute sig,Δ in ( Ai , A , D ) via Deﬁnition 9; 5: 6: 7:
Δ
Δ
if (sig,Δ in ( Ai , A , D ) > 0) then Ai = Ai ∪ Ai ; end Δ for ∀ Ai ∈ A − Ai
,Δ 8: if γ( Ai , D ) = γ( A, D ) then compute sigout ( Ai , A, D ) via Deﬁnition 10; 9: end Δ Δ ,Δ 10: if sig,Δ out ( A h , A , D ) = max{ sig out ( A h , A , D )} then Ai = Ai ∪ A h ; 11: end 12: end Δ 13: for ∀ Ai ∈ Ai , Δ
Δ
Δ
Δ
14: if γ( Ai − Ai , D ) = γ( A, D ) then Ai = Ai − Ai ; 15: end 16: end Δ 17: return granularity reduction set Ai ; 18: end
Therefore, we can obtain two reductions by utilizing Algorithm 1. Example 1. This paper calculates the granularity importance of 10 online investment schemes given in Reference [12]. After comparing and analyzing the obtained granularity importance degree, we can obtain the reduction results of 5 evaluation sites through Algorithm 1, and the detailed calculation steps are as follows. According to [12], we can get A {{ x1 , x2 , x4 , x6 , x8 }, { x3 , x5 , x7 , x9 , x10 }}. (1)
=
{ A1 , A2 , A3 , A4 , A5 }, A
⊆
A, U/D
=
Reduction set of OMRS
First of all, we can calculate the internal importance degree of OMRS by Theorem 2 as shown in Table 2. Table 2. Internal importance degree of optimistic multigranulation rough sets (OMRS). A1 0 sigO in ( Ai , A , D ) 0.025 sig,O in ( Ai , A , D )
A2
A3
A4
A5
0.15 0.375
0.05 0.225
0 0
0.05 0
Then, according to Algorithm 1, we can deduce the initial granularity set is { A1 , A2 , A3 }. Inspired by Deﬁnition 5, we obtain rO ({ A2 , A3 }, D ) = rO ( A, D ) = 1. So, the reduction set of the OMRS is AO i = { A2 , A3 }. As shown in Table 2, when using the new method to calculate internal importance degree, more discriminative granularities can be generated, which are more convenient for screening out the required granularities. In literature [12], the approximate quality of granularity A2 in the reduction set is different from that of the whole granularity set, so it is necessary to calculate the external importance degree again. When calculating the internal and external importance degree, References [10,12] only considered the direct inﬂuence of the single granularity on the granularity A2 , so the inﬂuence of the granularity A2 on the overall decisionmaking can’t be fully reﬂected. 214
Symmetry 2018, 10, 662
(2)
Reduction set of IMRS
Similarly, by using Theorem 2, we can get the internal importance degree of each site under IMRS, as shown in Table 3. Table 3. Internal importance degree of pessimistic multigranulation rough sets (IMRS).
I ( A , A , D ) sigin i sig,Iin ( Ai , A, D )
A1
A2
A3
A4
A5
0 0
0.05 0.025
0 0
0 0.025
0 0.025
According to Algorithm 1, the sites 2, 4, and 5 with internal importance degrees greater than 0, which are added to the granularity reduction set as the initial granularity set, and then the approximate quality of it can be calculated as follows: r I ({ A2 , A4 }, D ) = r I ({ A4 , A5 }, D ) = r I ( A, D ) = 0.2. Namely, the reduction set of IMRS is AiI = { A2 , A4 } or AiI = { A4 , A5 } without calculating the external importance degree. In this paper, when calculating the internal and external importance degree of each granularity, the inﬂuence of removing other granularities on decisionmaking is also considered. According to Theorem 2, after calculating the internal importance degree of OMRS and IMRS, if the approximate quality of each granularity in the reduction sets are the same as the overall granularities, it is not necessary to calculate the external importance degree again, which can reduce the amount of computation. 4. Novel MultiGranulation Rough Intuitionistic Fuzzy Sets Models In Example 1, two reduction sets are obtained under IMRS, so we need a novel method to obtain more accurate granularity reduction results by calculating granularity reduction. In order to obtain the optimal determined site selection result, we combine the optimistic and pessimistic multigranulation reduction sets based on Algorithm 1 with IFS, respectively, and construct the following four new MRIFS models. Deﬁnition 11 ([22,25]). Suppose IS = (U, A, V, f ) is an information system, A = { A1 , A2 , · · · , Am }. ∀ E ⊆ U, E are IFS. Then the lower and upper approximations of optimistic MRIFS of Ai are respectively deﬁned by: O
m
∑ R Ai ( E) = {< x, μ m O
m
i =1
i
( E)
i =1y∈[ x ] A
O
O
( x ) >  x ∈ U }.
i =1
∑ RA
i
i =1
( x ) = ∧ sup μ E (y), ν m i =1y∈[ x ]
( x ) >  x ∈ U };
∑ R Ai ( E )
m
( x ) = ∨ inf μ E (y), ν m m
∑ R Ai ( E )
i =1
( x ), ν m
m
O
O
∑ R Ai ( E )
i =1
i =1
μm μm
O
∑ R Ai ( E )
where i =1
( x ), ν m
i =1
∑ R Ai ( E) = {< x, μ m
∑ RA
O
∑ R Ai ( E )
i =1
O i
( E)
i =1
i =1y∈[ x ]
Ai
m
O
∑ R Ai ( E )
Ai
( x ) = ∧ sup νE (y); ( x ) = ∨ inf νE (y). i =1y∈[ x ] A
i
where R Ai is an equivalence relation of x in A, [ x ] Ai is the equivalence class of R Ai ,and ∨ is a disjunction operation.
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Symmetry 2018, 10, 662
Deﬁnition 12 ([22,25]). Suppose IS =< U, A, V, f > is an information system, A = { A1 , A2 , · · · , Am }. ∀ E ⊆ U, E are IFS. Then the lower and upper approximations of pessimistic MRIFS of Ai can be described as follows: I
m
∑ R Ai ( E) = {< x, μ m
I
∑ R Ai ( E )
i =1
I
m
( x ), ν m
∑ R Ai ( E) = {< x, μ m
i =1
I
∑ R Ai ( E )
I
∑ R Ai ( E )
i =1
i =1
I
∑ R Ai ( E )
m
I
∑ R Ai ( E )
i
i =1
i =1y∈[ x ]
I
∑ R Ai ( E )
Ai
Ai
m
( x ) = ∨ sup μ E (y), ν m i =1y∈[ x ]
( x ) = ∨ sup νE (y);
i =1
m
μm
( x ) >  x ∈ U }.
∑ R Ai ( E )
( x ) = ∧ inf μ E (y), ν m i =1y∈[ x ] A
I
( x ), ν m
m
μm
( x ) >  x ∈ U };
i =1
i =1
where
I
∑ R Ai ( E))
i =1
( x ) = ∧ inf νE (y). i =1y∈[ x ] A
i =1
i
where [ x ] Ai is the equivalence class of x about the equivalence relation R Ai , and ∧ is a conjunction operation. Deﬁnition 13. Suppose IS =< U, A, V, f > is an information system, AO i = { A1 , A2 , · · · , Ar } ⊆ A, A = { A1 , A2 , · · · , Am }. And R Ai O is an equivalence relation of x with respect to the attribute reduction set AO i under OMRS, [ x ] Ai O is the equivalence class of R Ai O . Let E be IFS of U and they can be characterized by a pair of lower and upper approximations: O
r
∑ R AO ( E) = {< x, μ
i =1
i =1
O
r
∑ R AO ( E) = {< x, μ
i =1
μ
r
O
r
∑ R AO ( E )
i =1
μ
O
r
Ai O
i =1y∈[ x ]
( x ) >  x ∈ U };
r
O
( x ) >  x ∈ U }.
∑ R AO ( E) i
r
O
r
∑ R AO ( E)
i =1
Ai O
O
i
i =1
i
( x ) = ∧ sup μ E (y), ν
i
O
( x ), ν
r
∑ R AO ( E )
r
∑ R AO ( E)
i =1
inf μ E (y), ν
i =1y∈[ x ]
i
i =1
r
(x) = ∨
O
∑ R AO ( E)
i =1
( x ), ν
i
r
i
where
O
r
∑ R AO ( E)
i
i =1y∈[ x ]
i
r
O
r
∑ R AO ( E)
i =1
( x ) = ∧ sup νE (y);
(x) = ∨
inf νE (y).
i =1y∈[ x ]
i
Ai O
Ai O
O
r
If ∑ R AO ( E) = ∑ R AO ( E), then E can be called OOMRIFS. i =1
i =1
i
i
Deﬁnition 14. Suppose IS =< U, A, V, f > is an information system, ∀ E ⊆ U, E are IFS. AO i = { A1 , A2 , · · · , Ar } ⊆ A, A = { A1 , A2 , · · · , Am }. where AO i is an optimistic multigranulation attribute reduction set. Then the lower and upper approximations of pessimistic MRIFS under optimistic multigranulation environment can be deﬁned as follows: I
r
∑ R AO ( E) = {< x, μ
i =1
i
I
r
∑ R AO ( E) = {< x, μ
i =1
i
I
r
∑ R AO ( E )
i =1
I
r
∑ R AO ( E )
i =1
( x ), ν
r
( x ), ν
i
216
I
( x ) >  x ∈ U };
I
( x ) >  x ∈ U }.
∑ R AO ( E )
i =1
i
i
r
∑ R AO ( E )
i =1
i
Symmetry 2018, 10, 662
where μ
r
I
r
∑ R AO ( E )
i =1
μ
Ai O
∑ R AO ( E )
( x ) = ∨ sup μ E (y), ν i =1y∈[ x ]
i
I
r
Ai O
I
r
∑ R AO ( E)
i =1
r
I
r
inf μ E (y), ν
i =1y∈[ x ]
i
i =1
r
(x) = ∧
i =1y∈[ x ]
i
r
I
r
∑ R AO ( E)
i =1
( x ) = ∨ sup νE (y); (x) = ∧
inf νE (y).
i =1y∈[ x ]
i
I
r
Ai O
Ai O
I
r
I
r
The pair ( ∑ R AO ( E), ∑ R AO ( E)) are called OIMRIFS, if ∑ R AO ( E) = ∑ R AO ( E). i =1
i =1
i
i =1
i
i =1
i
i
According to Deﬁnitions 13 and 14, the following theorem can be obtained. Theorem 4. Let IS =< U, A, V, f > be an information system, AO i = { A1 , A2 , · · · , Ar } ⊆ A, A = { A1 , A2 , · · · , Am }, and E1 , E2 be IFS on U. Comparing with Deﬁnitions 13 and 14, the following proposition is obtained. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
O
r
r
∑ R AO ( E1 ) = ∪ R AO O ( E1 );
i =1
i =1
i
O
r
i
r
O
∑ R AO ( E1 ) = ∩ R AO ( E1 );
i =1 r
i
i =1
i
i =1
I
i
r
∑ R AO ( E1 ) = ∩ R AO I ( E1 ); i =1
I
r
i
r
∑ R AO ( E1 ) = ∪ R AO I ( E1 );
i =1
i =1
i
I
r
i
O
r
∑ R AO ( E1 ) ⊆ ∑ R AO ( E1 );
i =1
i =1
i
O
r
i
I
r
∑ R AO ( E1 ) ⊆ ∑ R AO ( E1 );
i =1
i
r
O
i =1
i
O
r
O
r
I
r
I
r
I
r
∑ R AO ( E1 ∩ E2 ) = ∑ R AO ( E1 ) ∩ ∑ R AO ( E2 ), ∑ R AO ( E1 ∩ E2 ) = ∑ R AO ( E1 ) ∩ ∑ R AO ( E2 );
i =1
i =1
i
O
r
i =1
i
O
r
i =1
i
O
r
i =1
i
I
r
i =1
i
I
r
i
I
r
∑ R AO ( E1 ∪ E2 ) = ∑ R AO ( E1 ) ∪ ∑ R AO ( E2 ), ∑ R AO ( E1 ∪ E2 ) = ∑ R AO ( E1 ) ∪ ∑ R AO ( E2 );
i =1 r
i
i =1
i
O
i =1 r
i
i =1
i
O
i =1 r
i
i =1
i
O
i =1 r
i
i =1
i
I
i =1 r
i
i =1
i
I
i =1 r
i
i =1
i
I
∑ R AO ( E1 ∪ E2 ) ⊇ ∑ R AO ( E1 ) ∪ ∑ R AO ( E2 ), ∑ R AO ( E1 ∪ E2 ) ⊇ ∑ R AO ( E1 ) ∪ ∑ R AO ( E2 ); O
r
O
r
O
r
I
r
I
r
I
r
∑ R AO ( E1 ∩ E2 ) ⊆ ∑ R AO ( E1 ) ∩ ∑ R AO ( E2 ), ∑ R AO ( E1 ∩ E2 ) ⊆ ∑ R AO ( E1 ) ∩ ∑ R AO ( E2 ).
i =1
i =1
i
i
i =1
i =1
i
i =1
i
i
i =1
i
Proof. It is easy to prove by the Deﬁnitions 13 and 14. Deﬁnition 15. Let IS =< U, A, V, f > be an information system, and E be IFS on U. AiI = { A1 , A2 , · · · , Ar } ⊆ A, A = { A1 , A2 , · · · , Am }, where AiI is a pessimistic multigranulation attribute reduction set. Then, the pessimistic optimistic lower and upper approximations of E with respect to equivalence relation R Ai I are deﬁned by the following formulas: O
r
∑ R A I ( E) = {< x, μ
i =1
i
O
r
∑ R A I ( E) = {< x, μ
i =1
i
O
r
∑ R A I ( E)
i =1
O
r
∑ R A I ( E)
i =1
( x ), ν
r
( x ), ν
i
217
O
( x ) >  x ∈ U };
O
( x ) >  x ∈ U }.
∑ R A I ( E)
i =1
i
i
r
∑ R A I ( E)
i =1
i
Symmetry 2018, 10, 662
where
r
μ
O
r
∑ R A I ( E)
i =1
μ
∑ R A I ( E)
i =1y∈[ x ]
r
O
r
∑ R A I ( E)
i =1
( x ) = ∧ sup μ E (y), ν
i
O
Ai I
r
O
r
inf μ E (y), ν
i =1y∈[ x ]
i
i =1
r
(x) = ∨
Ai I
( x ) = ∧ sup νE (y); i =1y∈[ x ]
i
r
O
r
∑ R A I ( E)
i =1
(x) = ∨
Ai I
inf νE (y).
i =1y∈[ x ]
i
Ai I
O
r
If ∑ R A I ( E) = ∑ R A I ( E), then E can be called IOMRIFS. i =1
i =1
i
i
Deﬁnition 16. Let IS =< U, A, V, f > be an information system, and E be IFS on U. AiI = { A1 , A2 , · · · , Ar } ⊆ A, A = { A1 , A2 , · · · , Am }, where AiI is a pessimistic multigranulation attribute reduction set. Then, the pessimistic lower and upper approximations of E under IMRS are deﬁned by the following formulas: I
r
∑ R A I ( E) = {< x, μ
i =1
i
I
r
∑ R A I ( E) = {< x, μ
i =1
i
r
I
r
∑ R A I ( E)
i =1
μ
(x) = ∧
I
r
( x ), ν
Ai I
i =1y∈[ x ]
Ai I
r
I
( x ) >  x ∈ U }.
∑ R A I ( E) i
r
I
r
∑ R A I ( E)
i =1
( x ) = ∨ sup μ E (y), ν
i
( x ) >  x ∈ U };
i
i =1
i
r
∑ R A I ( E)
i =1
I
r
∑ R A I ( E)
I
r
∑ R A I ( E)
i =1
inf μ E (y), ν
i =1y∈[ x ]
i
( x ), ν
i
i =1
where μ
I
r
∑ R A I ( E)
i =1
i =1y∈[ x ]
i
r
I
r
∑ R A I ( E)
i =1
( x ) = ∨ sup νE (y); (x) = ∧
Ai I
inf νE (y).
i =1y∈[ x ]
i
Ai I
where R Ai I is an equivalence relation of x about the attribute reduction set AiI under IMRS, [ x ] Ai O is the equivalence class of R Ai I . I
r
I
r
I
r
I
r
If ∑ R A I ( E) = ∑ R A I ( E), then the pair ( ∑ R A I ( E), ∑ R A I ( E)) is said to be IIMRIFS. i =1
i
i =1
i =1
i
i
i =1
i
According to Deﬁnitions 15 and 16, the following theorem can be captured. Theorem 5. Let IS =< U, A, V, f > be an information system, AiI = { A1 , A2 , · · · , Ar } ⊆ A, A = { A1 , A2 , · · · , Am }, and E1 , E2 be IFS on U. Then IOMRIFS and IIOMRIFS models have the following properties: (1) (2) (3) (4) (5) (6) (7) (8)
O
r
r
∑ R A I ( E1 ) = ∪ R A I O ( E1 );
i =1
i =1
i
O
r
i
r
O
∑ R A I ( E1 ) = ∩ R A I ( E1 );
i =1 r
i
i =1
i
i =1
I
i
r
∑ R A I ( E1 ) = ∪ R A I I ( E1 ); i =1
I
r
i
r
∑ R A I ( E1 ) = ∪ R A I I ( E1 );
i =1
i =1
i
I
r
i
O
r
∑ R A I ( E1 ) ⊆ ∑ R A I ( E1 );
i =1
i =1
i
O
r
i
I
r
∑ R A I ( E1 ) ⊆ ∑ R A I ( E1 ).
i =1
i
r
O
i =1
i
O
r
O
r
I
r
I
r
I
r
∑ R A I ( E1 ∩ E2 ) = ∑ R A I ( E1 ) ∩ ∑ R A I ( E2 ), ∑ R A I ( E1 ∩ E2 ) = ∑ R A I ( E1 ) ∩ ∑ R A I ( E2 );
i =1
i =1
i
O
r
i =1
i
O
r
i =1
i
O
r
i =1
i
I
r
i =1
i
I
r
i
I
r
∑ R A I ( E1 ∪ E2 ) = ∑ R A I ( E1 ) ∪ ∑ R A I ( E2 ), ∑ R A I ( E1 ∪ E2 ) = ∑ R A I ( E1 ) ∪ ∑ R A I ( E2 );
i =1
i
i =1
i
i =1
i
i =1
i
218
i =1
i
i =1
i
Symmetry 2018, 10, 662
(9) (10)
O
r
O
r
O
r
I
r
I
r
I
r
∑ R A I ( E1 ∪ E2 ) ⊇ ∑ R A I ( E1 ) ∪ ∑ R A I ( E2 ), ∑ R A I ( E1 ∪ E2 ) ⊇ ∑ R A I ( E1 ) ∪ ∑ R A I ( E2 );
i =1
i =1
i
O
r
i =1
i
O
r
i =1
i
O
r
i =1
i
I
r
i =1
i
I
r
i
I
r
∑ R A I ( E1 ∩ E2 ) ⊆ ∑ R A I ( E1 ) ∩ ∑ R A I ( E2 ), ∑ R A I ( E1 ∩ E2 ) ⊆ ∑ R A I ( E1 ) ∩ ∑ R A I ( E2 ).
i =1
i =1
i
i =1
i
i
i =1
i =1
i
i
i =1
i
Proof. It can be derived directly from Deﬁnitions 15 and 16. The characteristics of the proposed four models are further veriﬁed by Example 2 below. Example 2. (Continued with Example 1). From Example 1, we know that these 5 sites are evaluated by 10 investment schemes respectively. Suppose they have the following IFS with respect to 10 investment schemes #
E=
[0.25,0.43] [0.51,0.28] [0.54,0.38] [0.37,0.59] [0.49,0.35] [0.92,0.04] [0.09,0.86] [0.15,0.46] , , , , , , , , x1 x2 x3 x4 x5 x6 x7 x8
[0.72,0.12] [0.67,0.23] , x x9 10
$
.
(1) In OOMRIFS, the lower and upper approximations of OOMRIFS can be calculated as follows: O
r
∑ R AO ( E ) =
i =1
i
O
r
∑ R AO ( E ) =
i =1
#
[0.25,0.59] [0.49,0.38] [0.49,0.38] [0.25,0.59] [0.49,0.38] [0.25,0.46] [0.09,0.86] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.15,0.46] [0.15,0.46] [0.67,0.23] , , x x8 x9 10
#
i
$
,
[0.51,0.28] [0.51,0.28] [0.54,0.35] [0.51,0.28] [0.54,0.35] [0.92,0.04] [0.54,0.35] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.15,0.46] [0.72,0.12] [0.67,0.23] , , x x8 x9 10
$
.
(2) Similarly, in OIMRIFS, we have: I
r
∑ R AO ( E ) =
i =1
i
I
r
∑ R AO ( E ) =
i =1
#
i
[0.25,0.59] [0.25,0.59] [0.09,0.86] [0.25,0.59] [0.09,0.86] [0.15,0.59] [0.09,0.86] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.15,0.46] [0.09,0.86] [0.09,0.86] , , x x8 x9 10
#
$
,
[0.92,0.04] [0.54,0.28] [0.54,0.28] [0.92,0.04] [0.54,0.28] [0.92,0.04] [0.72,0.12] , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.92,0.04] [0.92,0.04] [0.72,0.12] , , x x8 x9 10
$
.
From the above results, Figure 1 can be drawn as follows:
Figure 1. The lower and upper approximations of OOMRIFS and OIMRIFS.
Note that μ1 = μOO ( x j ) and ν1 = νOO ( x j ) represent the lower approximation of OOMRIFS; μ2 = μOO ( x j ) and ν2 = νOO ( x j ) represent the upper approximation of OOMRIFS; 219
,
Symmetry 2018, 10, 662
μ3 = μOI ( x j ) and ν3 = νOI ( x j ) represent the lower approximation of OIMRIFS; μ4 = μOI ( x j ) and ν4 = νOI ( x j ) represent the upper approximation of OIMRIFS. Regarding Figure 1, we can get, μOI ( x j ) ≥ μOO ( x j ) ≥ μOO ( x j ) ≥ μOI ( x j ); νOI ( x j ) ≥ νOO ( x j ) ≥ νOO ( x j ) ≥ νOI ( x j ). As shown in Figure 1, the rules of Theorem 4 are satisﬁed. By constructing the OOMRIFS and OIMRIFS models, we can reduce the subjective scoring errors of experts under intuitionistic fuzzy conditions. (3) Similar to (1), in IOMRIFS, we have: O
r
∑ R A I ( E) =
i =1
i
[0.25,0.43] [0.25,0.43] [0.25,0.43] [0.37,0.59] [0.25,0.43] [0.25,0.46] [0.09,0.86] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.15,0.46] [0.67,0.23] [0.67,0.23] , , x x8 x9 10 O
r
∑ R A I ( E) =
i =1
#
#
i
$
,
[0.51,0.28] [0.51,0.28] [0.54,0.35] [0.37,0.59] [0.49,0.35] [0.92,0.04] [0.51,0.35] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.49,0.35] [0.72,0.12] [0.67,0.23] , , x x8 x9 10
$
.
(4) The same as (1), in IIMRIFS, we can get: I
r
∑ R A I ( E) =
i =1
i
[0.25,0.59] [0.09,0.86] [0.09,0.86] [0.25,0.59] [0.09,0.86] [0.09,0.86] [0.09,0.86] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.09,0.86] [0.15,0.46] [0.67,0.23] , , x x8 x9 10 I
r
∑ R A I ( E) =
i =1
#
i
#
$
,
[0.92,0.04] [0.54,0.28] [0.92,0.04] [0.92,0.04] [0.54,0.28] [0.92,0.04] [0.92,0.04] , , , , , , , x1 x2 x3 x4 x5 x6 x7
[0.92,0.04] [0.92,0.04] [0.72,0.12] , , x x8 x9 10
$
.
From (3) and (4), we can obtain Figure 2 as shown:
Figure 2. The lower and upper approximations of IOMRIFS and IIMRIFS.
Note that μ5 = μ IO ( x j ) and ν5 = ν IO ( x j ) represent the lower approximation of IOMRIFS; μ6 = μ IO ( x j ) and ν6 = ν IO ( x j ) represent the upper approximation of IOMRIFS; μ7 = μ I I ( x j ) and ν7 = ν I I ( x j ) represent the lower approximation of IIMRIFS; μ8 = μ I I ( x j ) and ν8 = ν I I ( x j ) represent the upper approximation of IIMRIFS. For Figure 2, we can get, μ I I ( x j ) ≥ μ IO ( x j ) ≥ μ IO ( x j ) ≥ μ I I ( x j ); ν I I ( x j ) ≥ ν IO ( x j ) ≥ ν IO ( x j ) ≥ ν I I ( x j ).
220
Symmetry 2018, 10, 662
As shown in Figure 2, the rules of Theorem 5 are satisﬁed. Through the Example 2, we can obtain four relatively more objective MRIFS models, which are beneﬁcial to reduce subjective errors. 5. ThreeWay Decisions Models Based on MRIFS and Optimal Granularity Selection In order to obtain the optimal granularity selection results in the case of optimistic and pessimistic multigranulation sets, it is necessary to further distinguish the importance degree of each granularity in the reduction sets. We respectively combine the four MRIFS models mentioned above with threeway decisions theory to get four new threeway decisions models. By extracting the rules, the redundant objects in the reduction sets are removed, and the decision error is further reduced. Then the optimal granularity selection results in two cases are obtained respectively by constructing the comprehensive score function and comprehensive accuracy function measurement formulas of each granularity of the reduction sets. 5.1. ThreeWay Decisions Model Based on OOMRIFS Suppose AO i is the reduction set under OMRS. According to reference [46], the expected loss function ROO (ω∗ [ x ] AO )(∗ = P, B, N ) of object x can be obtained: i
ROO (ω P [ x ] AO ) = λ PP · μOO ( x ) + λ PN · νOO ( x ) + λ PB · πOO ( x ); i
ROO (ω N [ x ] AO ) = λ NP · μOO ( x ) + λ NN · νOO ( x ) + λ NB · πOO ( x ); i
ROO (ω B [ x ] AO ) = λ BP · μOO ( x ) + λ BN · νOO ( x ) + λ BB · πOO ( x ). i
where μOO ( x ) = μ r
∑ R O i =1 A i
r
O
( E)
(x) = ∨
inf
i =1y∈[ x ]
Ai O
μ E (y), νOO ( x ) = ν r
∑ R O i =1 A i
r
O
( E)
(x) = ∧
sup
i =1y∈[ x ]
νE (y), πOO ( x ) = 1 − μ r
∑ R O i =1 A i
Ai O
O
( E)
(x) − ν
( x ); O r ∑ R O ( E) i =1 A i
or μOO ( x ) = μ
r
O r ∑ R O ( E) i =1 A i
(x) = ∧
sup
i =1y∈[ x ]
μ E (y), νOO ( x ) = ν
r
O r ∑ R O ( E) i =1 A i
Ai O
(x) = ∨
inf
i =1y∈[ x ]
Ai O
νE (y), πOO ( x ) = 1 − μ
O r ∑ R O ( E) i =1 A i
(x) − ν
O r ∑ R O ( E) i =1 A i
( x ).
The minimumrisk decision rules derived from the Bayesian decision process are as follows: P : If R (ω P [ x ] AO ) ≤ R (ω B [ x ] AO ) and R (ω P [ x ] AO ) ≤ R (ω N [ x ] AO ), then x ∈ POS( X ); i
i
i
i
(N): If R (ω N [ x ] AO ) ≤ R (ω P [ x ] AO ) and R (ω N [ x ] AO ) ≤ R (ω B [ x ] AO ), then x ∈ NEG ( X ); i i i i (B): If R (ω B [ x ] AO ) ≤ R (ω N [ x ] AO ) and R (ω B [ x ] AO ) ≤ R (ω P [ x ] AO ), then x ∈ BND ( X ). i i i i Thus, the decision rules (P)(B) can be reexpressed concisely as: (P) rule satisﬁes: (μOO ( x ) ≤ (1 − πOO ( x )) ·
λ NN − λ PN λ BN − λ PN ) ∧ (μOO ( x ) ≤ (1 − πOO ( x )) · ); (λ PP − λ NP ) + (λ PN − λ NN ) (λ PP − λ BP ) + (λ PN − λ BN )
(N) rule satisﬁes: (μOO ( x ) < (1 − πOO ( x )) ·
λ PN − λ NN λ BN − λ NN ) ∧ (μOO ( x ) < (1 − πOO ( x )) · ); (λ NP − λ PP ) + (λ PN − λ NN ) (λ NP − λ BP ) + (λ BN − λ NN )
(B) rule satisﬁes: (μOO ( x ) > (1 − πOO ( x )) ·
λ BN − λ PN λ BN − λ NN ) ∧ (μOO ( x ) ≥ (1 − πOO ( x )) · ). (λ PN − λ BN ) + (λ BP − λ PP ) (λ BN − λ NN ) + (λ NP − λ BP )
Therefore, the threeway decisions rules based on OOMRIFS are as follows: (P1): If μOO ( x ) ≥ (1 − πOO ( x )) · α, then x ∈ POS( X ); (N1): If μOO ( x ) ≤ (1 − πOO ( x )) · β, then x ∈ NEG ( X ); (B1): If (1 − πOO ( x )) · β ≤ μOO ( x ) and μOO ( x ) ≤ (1 − πOO ( x )) · α, then x ∈ BND ( X ).
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Symmetry 2018, 10, 662
5.2. ThreeWay Decisions Model Based on OIMRIFS Suppose AO i is the reduction set under OMRS. According to reference [46], the expected loss functions ROO (ω∗ [ x ] AO )(∗ = P, B, N ) of an object x are presented as follows: i
ROI (ω
P [ x ] AO ) i
= λ PP · μOI ( x ) + λ PN · νOI ( x ) + λ PB · πOI ( x );
ROI (ω N [ x ] AO ) = λ NP · μOI ( x ) + λ NN · νOI ( x ) + λ NB · πOI ( x ); i
ROI (ω B [ x ] AO ) = λ BP · μOI ( x ) + λ BN · νOI ( x ) + λ BB · πOI ( x ). i
where μOI ( x ) = μ
(x) I r ∑ R O ( E) i =1 A i
r
= ∧
inf
i =1y∈[ x ]
μ E (y), νOI ( x ) = v
Ai O
(x) I r ∑ R O ( E) i =1 A i
r
= ∨
sup v E (y), πOI ( x ) = 1 − μ
i =1y∈[ x ]
Ai O
(x) I r ∑ R O ( E) i =1 A i
−v
( x ); I r ∑ R O ( E) i =1 A i
(x) − ν
( x ).
or μOI ( x ) = μ
r
I r ∑ R O ( E) i =1 A i
(x) = ∨
r
sup μ E (y), νOI ( x ) = ν
i =1y∈[ x ]
I r ∑ R O ( E) i =1 A i
Ai O
(x) = ∧
inf
i =1y∈[ x ]
νE (y), πOI ( x ) = 1 − μ
Ai O
I r ∑ R O ( E) i =1 A i
I r ∑ R O ( E) i =1 A i
Therefore, the threeway decisions rules based on OIMRIFS are as follows: (P2): If μOI ( x ) ≥ (1 − πOI ( x )) · α, then x ∈ POS( X ); (N2): If μOI ( x ) ≤ (1 − πOI ( x )) · β, then x ∈ NEG ( X ); (B2): If (1 − πOI ( x )) · β ≤ μOI ( x ) and μOI ( x ) ≤ (1 − πOI ( x )) · α, then x ∈ BND ( X ). 5.3. ThreeWay Decisions Model Based on IOMRIFS Suppose AiI is the reduction set under IMRS. According to reference [46], the expected loss functions R IO (ω∗ [ x ] A I )(∗ = P, B, N ) of an object x are as follows: i
R IO (ω
P [ x ] A I ) i
= λ PP · μ IO ( x ) + λ PN · ν IO ( x ) + λ PB · π IO ( x );
R IO (ω N [ x ] A I ) = λ NP · μ IO ( x ) + λ NN · ν IO ( x ) + λ NB · π IO ( x ); i
R IO (ω B [ x ] A I ) = λ BP · μ IO ( x ) + λ BN · ν IO ( x ) + λ BB · π IO ( x ). i
where μ IO ( x ) = μ
(x) O r ∑ R I ( E) i =1 A i
r
= ∨
inf μ E (y), ν IO ( x ) = ν
i =1y∈[ x ]
(x) O r ∑ R I ( E) i =1 A i
Ai I
r
= ∧ sup νE (y), π IO ( x ) = 1 − μ i =1y∈[ x ]
Ai I
(x) O r ∑ R I ( E) i =1 A i
−ν
( x ); O r ∑ R I ( E) i =1 A i
or μ IO ( x ) = μ
r
O r ∑ R I ( E) i =1 A i
( x ) = ∧ sup μ E (y), ν IO ( x ) = ν i =1y∈[ x ]
r
O r ∑ R I ( E) i =1 A i
Ai I
(x) = ∨
inf νE (y), π IO ( x ) = 1 − μ
i =1y∈[ x ]
Ai I
O r ∑ R I ( E) i =1 A i
(x) − ν
O r ∑ R I ( E) i =1 A i
( x ).
Therefore, the threeway decisions rules based on IOMRIFS are as follows: (P3): If μ IO ( x ) ≥ (1 − π IO ( x )) · α, then x ∈ POS( X ); (N3): If μ IO ( x ) ≤ (1 − π IO ( x )) · β, then x ∈ NEG ( X ); (B3): If (1 − π IO ( x )) · β ≤ μ IO ( x ) and μ IO ( x ) ≤ (1 − π IO ( x )) · α, then x ∈ BND ( X ). 5.4. ThreeWay Decisions Model Based on IIMRIFS Suppose AiI is the reduction set under IMRS. Like Section 5.1, the expected loss functions ∗ [ x ] A I )(∗ = P, B, N ) of an object x are as follows:
R I I (ω
i
R I I (ω P [ x ] A I ) = λ PP · μ I I ( x ) + λ PN · ν I I ( x ) + λ PB · π I I ( x ); i
R (ω N [ x ] A I ) = λ NP · μ I I ( x ) + λ NN · ν I I ( x ) + λ NB · π I I ( x ); II
i
R (ω B [ x ] A I ) = λ BP · μ I I ( x ) + λ BN · ν I I ( x ) + λ BB · π I I ( x ). II
i
where
222
Symmetry 2018, 10, 662
μ I I (x) = μ
r
(x) I r ∑ R I ( E) i =1 A i
= ∧
inf μ E (y), ν I I ( x ) = ν
(x) I r ∑ R I ( E) i =1 A i
= ∨ sup μ E (y), ν I I ( x ) = ν
i =1y∈[ x ]
(x) I r ∑ R I ( E) i =1 A i
Ai I
r
= ∨ sup νE (y), π I I ( x ) = 1 − μ i =1y∈[ x ]
Ai I
(x) I r ∑ R I ( E) i =1 A i
−ν
( x ); I r ∑ R I ( E) i =1 A i
(x) I r ∑ R I ( E) i =1 A i
−ν
or μ I I (x) = μ
r
i =1y∈[ x ]
(x) I r ∑ R I ( E) i =1 A i
Ai I
r
= ∧
inf νE (y), π I I ( x ) = 1 − μ
i =1y∈[ x ]
Ai I
( x ). I r ∑ R I ( E) i =1 A i
Therefore, the threeway decisions rules based on IIMRIFS are captured as follows: (P4): If μ I I ( x ) ≥ (1 − π I I ( x )) · α, then x ∈ POS( X ); (N4): If μ I I ( x ) ≤ (1 − π I I ( x )) · β, then x ∈ NEG ( X ); (B4): If (1 − π I I ( x )) · β ≤ μ I I ( x ) and μ I I ( x ) ≤ (1 − π I I ( x )) · α, then x ∈ BND ( X ). By constructing the above three decision models, the redundant objects in the reduction sets can be removed, which is beneﬁcial to the optimal granular selection. 5.5. Comprehensive Measuring Methods of Granularity 5( f 1 ) = (μ 5 ( f 1 ), ν 5 ( f 1 )), f 1 ∈ U, then the score Deﬁnition 17 ([40]). Let an intuitionistic fuzzy number E E E 5 function of E( f 1 ) is calculated as: 5( f 1 )) = μ 5 ( f 1 ) − ν 5 ( f 1 ). S( E E E 5( f 1 ) is deﬁned as: The accuracy function of E 5( f 1 )) = μ 5 ( f 1 ) + ν 5 ( f 1 ). H (E E E 5( f 1 )) ≤ 1 and 0 ≤ H ( E 5( f 1 )) ≤ 1. where −1 ≤ S( E Deﬁnition 18. Let DIS = (U, C ∪ D ) be a decision information system, A = { A1 , A2 , · · · , Am } are m subattributes of C. Suppose E are IFS on the universe U = { x1 , x2 , · · · , xn }, deﬁned by μ Ai ( x j ) and νAi ( x j ), where μ Ai ( x j ) and νAi ( x j ) are their membership and nonmembership functions respectively. [ x j ] A  is the i number of equivalence classes of xj on granularity Ai , U/D = { X1 , X2 , · · · , Xs } is the partition induced by the decision attributes D. Then, the comprehensive score function of granularity Ai is captured as: CSFAi ( E) =
n 1 × s j=1,n∑ ∈[ x ]
j A i
μ Ai ( x j ) − νAi ( x j ) . [ x j ] A  i
The comprehensive accuracy function of granularity Ai is captured as: CAFAi ( E) =
n 1 × s j=1,n∑ ∈[ x ]
j A i
μ Ai ( x j ) + νAi ( x j ) . [ x j ] A  i
where −1 ≤ CSFAi ( E) ≤ 1 and 0 ≤ CAFAi ( E) ≤ 1. With respect to Deﬁnition 19, according to references [27,39], we can deduce the following rules. Deﬁnition 19. Let two granularities A1 , A2 , then we have: (1) (2) (3)
If CSFA1 ( E) > CSFA2 ( E), then A2 is smaller than A1 , expressed as A1 > A2 ; If CSFA1 ( E) < CSFA2 ( E), then A1 is smaller than A2 , expressed as A1 < A2 ; If CSFA1 ( E) = CSFA2 ( E), then (i) (ii) (iii)
If CSFA1 ( E) = CSFA2 ( E), then A2 is equal to A1 , expressed as A1 = A2 ; If CSFA1 ( E) > CSFA2 ( E), then A2 is smaller than A1 , expressed as A1 > A2 ; If CSFA1 ( E) < CSFA2 ( E), then A1 is smaller than A2 , expressed as A1 < A2 .
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Symmetry 2018, 10, 662
5.6. Optimal Granularity Selection Algorithm to Derive ThreeWay Decisions from MRIFS I Suppose the reduction sets of optimistic and IMRS are AO i and Ai respectively. In this section, we take the reduction set under OMRS as an example to make the result AO i of optimal granularity selection.
Algorithm 2. Optimal granularity selection algorithm to derive threeway decisions from MRIFS Input: DIS = (U, C ∪ D, V, f ), A = { A1 , A2 , · · · , Am } be m subattributes of condition attributes C, ∀ Ai ∈ A , U/D = { X1 , X2 , · · · , Xs }, IFS E; Output: Optimal granularity selection result AO i . 1: compute via Algorithm 1; 2: if  AO i >1 3: for ∀ Ai ∈ AO i 4: compute μ r ( x j ), ν r ( x j ), μ r ( x j ) and ν r ( x j ); Δ Δ Δ Δ ∑ R AO ( E)
i =1
i
∑ R AO ( E)
i =1
∑ R AO ( E)
i
i =1
i
∑ R AO ( E)
i =1
i
5: according (P1)(B1) and (P2)(B2), compute POS( XOΔ ), NEG ( XOΔ ), BND ( XOΔ ), POS( XOΔ ), NEG ( XOΔ ), BND ( XOΔ ); 6: if NEG ( XOΔ ) = U or NEG ( XOΔ ) = U OΔ compute U/AOΔ i , CSFAOΔ ( E ), CAFAOΔ ( E ) or (U/Ai ), (CSFAOΔ ( E ), CAFAOΔ ( E );
7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:
i
i
i
according to Deﬁnition 19 to get AO i ; O return Ai = Ai ;
i
end else return NULL; end end end else O return AO i = Ai ; end
6. Example Analysis 3 (Continued with Example 2) In Example 1, only site 1 can be ignored under optimistic and pessimistic multigranulation conditions, so it can be determined that site 1 does not need to be evaluated, while sites 2 and 3 need to be further investigated under the environment of optimistic multigranulation. At the same time, with respect to the environment of pessimistic multigranulation, comprehensive considera tion site 3 can ignore the assessment and sites 2, 4 and 5 need to be further investigated. According to Example 1, we can get that the reduction set of OMRS is { A2 , A3 }, but in the case of IMRS, there are two reduction sets, which are contradictory. Therefore, two reduction sets should be reconsidered simultaneously, so the joint reduction set under IMRS is { A2 , A4 , A5 }. Where the corresponding granularity structures of sites 2, 3, 4 and 5 are divided as follows: U/A2 U/A3 U/A4 U/A5
= {{ x1 , x2 , x4 }, { x3 , x5 , x7 }, { x6 , x8 , x9 }, { x10 }}, = {{ x1 , x4 , x6 }, { x2 , x3 , x5 }, { x8 }, { x7 , x9 , x10 }}, = {{ x1 , x2 , x3 , x5 }, { x4 }, { x6 , x7 , x8 }, { x9 , x10 }}, = {{ x1 , x3 , x4 , x6 }, { x2 , x7 }, { x5 , x8 }, { x9 , x10 }}.
According to reference [11], we can get: −0 −2 = 0.75; β = (2−02)+( = 0.33. α = (8−28)+( 2−0) 6−2) The optimal site selection process under optimistic and IMRS is as follows: (1) Optimal site selection based on OOMRIFS 224
Symmetry 2018, 10, 662
According to the Example 2, we can get the values of evaluation functions μOO ( x j ), (1 − πOO ( x j )) · α, (1 − πOO ( x j )) · β, μOO ( x j ), (1 − πOO ( x j )) · α and (1 − πOO ( x j )) · β of OOMRIFS, as shown in Table 4. Table 4. The values of evaluation functions for OOMRIFS.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
μOO (xj )
(1−π OO (xj ))·α
0.25 0.49 0.49 0.25 0.49 0.25 0.09 0.15 0.15 0.67
0.63 0.6525 0.6525 0.63 0.6525 0.5325 0.7125 0.4575 0.4575 0.675
(1−π OO (xj ))· β
μOO (xj )
(1−π OO (xj ))·α
(1−π OO (xj ))· β
0.51 0.51 0.54 0.51 0.54 0.92 0.54 0.15 0.72 0.67
0.5925 0.5925 0.6675 0.5925 0.6675 0.72 0.6675 0.4575 0.63 0.675
0.2607 0.2607 0.2937 0.2607 0.2937 0.3168 0.2937 0.2013 0.2772 0.297
0.2772 0.2871 0.2871 0.2772 0.2871 0.2343 0.3135 0.2013 0.2013 0.297
We can get decision results of the lower and upper approximations of OOMRIFS by threeway decisions of the Section 5.1, as follows: POS( XOO ) = φ, NEG ( XOO ) = { x1 , x4 , x7 , x8 , x9 }, BND ( XOO ) = { x2 , x3 , x5 , x6 , x10 }; POS( XOO ) = { x6 , x9 }, NEG ( XOO ) = { x8 }, BND ( XOO ) = { x2 , x3 , x5 }. In the light of threeway decisions rules based on OOMRIFS, after getting rid of the objects in the rejection domain, we choose to fuse the objects in the delay domain with those in the acceptance domain for the optimal granularity selection. Therefore, the new granularities A2 , A3 are as follows: U/AOI 2 = {{ x2 }, { x3 , x5 }, { x6 }, { x10 }}, U/AOI 3 = {{ x2 , x3 , x5 }, { x6 }, { x10 }};
U/AOI 2 = {{ x1 , x2 , x4 }, { x3 , x5 , x7 }, { x6 , x9 }, { x10 }},
U/AOI 3 = {{ x1 , x4 , x6 }, { x2 , x3 , x5 }, { x7 , x9 , x10 }}. Then, according to Deﬁnition 18, we can get: CSFAOO ( E) 2
= =
1 s 1 4
× ×
n
μ Ai ( x j )−νAi ( x j ) [ x j ] A 
∑
j=1,n∈[ x j ] A
i
i
μ AOO ( x j )−νAOO ( x j )
10
2
∑
j=1,n∈[ x j ] AOO
2
2
=
1 4
2
[ x j ] AOO 
× ((0.49 − 0.38) +
(0.49−0.38)+(0.49−0.38) 2
+ (0.25 − 0.46) + (0.67 − 0.23))
= 0.1125, CSFAOO ( E) 3
= =
1 s 1 3
× ×
n
μ Ai ( x j )−νAi ( x j ) [ x j ] A 
∑
j=1,n∈[ x j ] A
i
10
i
μ AOO ( x j )−νAOO ( x j ) 3
∑
j=1,n∈[ x j ] AOO 3
=
1 3
× ((0.25 − 0.46) +
3
[ x j ] AOO  3
(0.49−0.38)+(0.49−0.38)+(0.49−0.38) 3
= 0.1133; Similarly, we have: CSF OO ( E) = 0.4, CSF A2
AOO 3
( E) = 0.3533. 225
+ (0.81 − 0.14))
Symmetry 2018, 10, 662
From the above results, in OOMRIFS, we can see that we can’t get the selection result of sites 2 and 3 only according to the comprehensive score function of granularities A2 and A3 . Therefore, we need to further calculate the comprehensive accuracies to get the results as follows: CAFAOO ( E) 2
= =
1 s 1 4
× ×
n
μ Ai ( x j )+νAi ( x j ) [ x j ] A 
∑
j=1,n∈[ x j ] A
i
i
μ AOO ( x j )+νAOO ( x j )
10
2
∑
j=1,n∈[ x j ] AOO
2
2
=
1 4
2
[ x j ] AOO 
× ((0.49 + 0.38) +
(0.49+0.38)+(0.49+0.38) 2
+ (0.25 + 0.46) + (0.67 + 0.23))
= 0.8375, CAFAOO ( E) 3
= =
1 s 1 3
× ×
n
μ Ai ( x j )+νAi ( x j ) [ x j ] A 
∑
j=1,n∈[ x j ] A
i
10
i
μ AOO ( x j )+νAOO ( x j ) 3
∑
3
3
=
1 3
3
[ x j ] AOO 
j=1,n∈[ x j ] AOO
× ((0.25 + 0.46) +
(0.49+0.38)+(0.49+0.38)+(0.49+0.38) 3
+ (0.81 + 0.14))
= 0.8267; Analogously, we have: CAF OO ( E) = 0.87, CAF
AOO 3
A2
( E) = 0.86.
Through calculation above, we know that the comprehensive accuracy of the granularity A3 is higher, so the site 3 is selected as the selection result. (2) Optimal site selection based on OIMRIFS The same as (1), we can get the values of evaluation functions μOI ( x j ), (1 − πOI ( x j )) · α, (1 − πOI ( x
j )) ·
β, μOI ( x j ), (1 − πOI ( x j )) · α and (1 − πOI ( x j )) · β of OIMRIFS listed in Table 5. Table 5. The values of evaluation functions for OIMRIFS.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
μOI (xj )
(1−π OI (xj ))·α
(1−π OI (xj ))· β
μOI (xj )
(1−π OI (xj ))·α
(1−π OI (xj ))· β
0.25 0.25 0.09 0.25 0.09 0.15 0.09 0.15 0.09 0.09
0.63 0.63 0.7125 0.63 0.7125 0.555 0.7125 0.4575 0.7125 0.7125
0.2772 0.2772 0.3135 0.2772 0.3135 0.2442 0.3135 0.2013 0.3135 0.3135
0.92 0.54 0.54 0.92 0.54 0.92 0.72 0.92 0.92 0.72
0.72 0.615 0.615 0.72 0.615 0.72 0.63 0.72 0.72 0.63
0.3168 0.2706 0.2706 0.3168 0.2706 0.3168 0.2772 0.3168 0.3168 0.2772
We can get decision results of the lower and upper approximations of OIMRIFS by threeway decisions in the Section 5.2, as follows: POS( XOI ) = φ, NEG ( XOI ) = U, BND ( XOI ) = φ; POS( XOI ) = { x1 , x4 , x6 , x7 , x8 , x9 , x10 }, NEG ( XOI ) = φ, BND ( XOI ) = { x2 , x3 , x5 }. Hence, in the upper approximations of OIMRIFS, the new granularities A2 , A3 are as follows:
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Symmetry 2018, 10, 662
U/AOI 2 = {{ x1 , x2 , x4 }, { x3 , x5 , x7 }, { x6 , x8 , x9 }, { x10 }},
U/AOI 3 = {{ x1 , x4 , x6 }, { x2 , x3 , x5 }, { x8 }, { x7 , x9 , x10 }}. According to Deﬁnition 18, we can calculate that CSFAOI ( E) = CSFAOI ( E) = 0; 2
3
CAFAOI ( E) = CAFAOI ( E) = 0; 2
CSF
AOI 2
CAF
3
( E) = 0.6317, CSFAOI ( E) = 0.6783; 3
AOI 2
( E) = 0.885, CAFAOI ( E) = 0.905. 3
In OIMRIFS, the comprehensive score and comprehensive accuracy of the granularity A3 are both higher than the granularity A2 . So, we choose site 3 as the evaluation site. In reality, we are more inclined to select the optimal granularity in the case of more stringent requirements. According to (1) and (2), we can ﬁnd that the granularity A3 is a better choice when the requirements are stricter in four cases of OMRS. Therefore, we choose site 3 as the optimal evaluation site. (3) Optimal site selection based on IOMRIFS Similar to (1), we can obtain the values of evaluation functions μ IO ( x j ), (1 − π IO ( x j )) · α,
(1 − π IO ( x j )) · β, μ IO ( x j ), (1 − π IO ( x j )) · α and (1 − π IO ( x j )) · β of IOMRIFS, as described in Table 6. Table 6. The values of evaluation functions for IOMRIFS.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
μIO (xj )
(1−π IO (xj ))·α
(1−π IO (xj ))· β
μIO (xj )
(1−π IO (xj ))·α
(1−π IO (xj ))· β
0.25 0.25 0.25 0.37 0.25 0.25 0.09 0.15 0.67 0.67
0.51 0.51 0.51 0.72 0.51 0.5325 0.7125 0.4575 0.675 0.675
0.2244 0.2244 0.2244 0.3168 0.2244 0.2343 0.3135 0.2013 0.297 0.297
0.51 0.51 0.54 0.37 0.49 0.92 0.51 0.49 0.72 0.67
0.5925 0.5925 0.6675 0.72 0.63 0.72 0.645 0.63 0.63 0.675
0.2607 0.2607 0.2937 0.3168 0.2772 0.3168 0.2838 0.2772 0.2772 0.297
We can get decision results of the lower and upper approximations of IOMRIFS by threeway decisions in the Section 5.3, as follows: POS( X IO ) = φ, NEG ( X IO ) = { x7 , x8 }, BND ( X IO ) = { x1 , x2 , x3 , x4 , x5 , x6 , x9 , x10 }; POS( X IO ) = { x6 , x9 }, NEG ( X IO ) = φ, BND ( X IO ) = { x1 , x2 , x3 , x4 , x5 , x7 , x8 , x10 }. Therefore, the granularities A2 , A4 , A5 can be rewritten as follows: U/A2IO = {{ x1 , x2 , x4 }, { x3 , x5 }, { x6 , x9 }, { x10 }}, U/A4IO = {{ x1 , x2 , x3 , x5 }, { x4 }, { x6 }, { x9 , x10 }}, U/A5IO = {{ x1 , x3 , x4 , x6 }, { x2 }, { x5 }, { x9 , x10 }}; U/A2IO = {{ x1 , x2 , x4 }, { x3 , x5 , x7 }, { x6 , x8 , x9 }, { x10 }},
U/A4IO = {{ x1 , x2 , x3 , x5 }, { x4 }, { x6 , x7 , x8 }, { x9 , x10 }},
U/A5IO = {{ x1 , x3 , x4 , x6 }, { x2 , x7 }, { x5 , x8 }, { x9 , x10 }}. According to Deﬁnition 18, one can see that the results are captured as follows: CSFA IO ( E) = 0.0454, CSFA IO ( E) = −0.0567, CSFA IO ( E) = −0.0294; 2
CSF
A2IO
5
4
( E) = 0.3058, CSFA IO ( E) = 0.2227, CSFA IO ( E) = 0.2813. 5
4
227
Symmetry 2018, 10, 662
In summary, the comprehensive score function of the granularity A2 is higher than the granularity A3 in IOMRIFS, so we choose site 2 as the result of granularity selection. (4) Optimal site selection based on IIMRIFS In the same way as (1), we can get the values of evaluation functions μ I I ( x j ), (1 − π I I ( x j )) · α,
(1 − π I I ( x j )) · β, μ I I ( x j ), (1 − π I I ( x j )) · α and (1 − π I I ( x j )) · β of IIMRIFS, as shown in Table 7. Table 7. The values of evaluation functions for IIMRIFS.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
μII (xj )
(1−π II (xj ))·α
(1−π II (xj ))· β
μII (xj )
(1−π II (xj ))·α
(1−π II (xj ))· β
0.25 0.09 0.09 0.25 0.09 0.09 0.09 0.09 0.15 0.67
0.63 0.7125 0.7125 0.63 0.7125 0.7125 0.7125 0.7125 0.4575 0.675
0.2772 0.3135 0.3135 0.2772 0.3135 0.3135 0.3135 0.3135 0.2013 0.297
0.92 0.54 0.92 0.92 0.54 0.92 0.92 0.92 0.92 0.72
0.72 0.615 0.72 0.72 0.615 0.72 0.72 0.72 0.72 0.63
0.3168 0.2706 0.3168 0.3168 0.2706 0.3168 0.3168 0.3168 0.3168 0.2772
We can get decision results of the lower and upper approximations of IIMRIFS by threeway decisions in the Section 5.4, as follows: POS( X I I ) = φ, NEG ( X I I ) = { x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 }, BND ( X I I ) = { x10 }; POS( X I I ) = { x1 , x3 , x4 , x6 , x7 , x8 , x9 , x10 }, NEG ( X I I ) = φ, BND ( X I I ) = { x2 , x5 }. Therefore, the granularity structures of A2 , A4 , A5 can be rewritten as follows: U/A2I I = U/A4I I = U/A5I I = { x10 }; U/A2I I = {{ x1 , x2 , x4 }, { x3 , x5 , x7 }, { x6 , x8 , x9 }, { x10 }},
U/A4I I = {{ x1 , x2 , x3 , x5 }, { x4 }, { x6 , x7 , x8 }, { x9 , x10 }},
U/A5I I = {{ x1 , x3 , x4 , x6 }, { x2 , x7 }, { x5 , x8 }, { x9 , x10 }}. According to Deﬁnition 18, one can see that the results are captured as follows: CSFA I I ( E) = CSFA I I ( E) = CSFA I I ( E) = 0.44; 2
5
4
CAFA I I ( E) = CAFA I I ( E) = CAFA I I ( E) = 0.9; 2
5
4
CSFA I I ( E) = 0.7067, CSFA I I ( E) = 0.7675, CSFA I I ( E) = 0.69; 2
5
4
CAFA I I ( E) = 0.9067, CAFA I I ( E) = 0.9275, CAFA I I ( E) = 0.91. 2
5
4
In IIMRIFS, the values of the comprehensive score and comprehensive accuracy of granularity A4 are higher than A2 and A5 , so site 4 is chosen as the evaluation site. Considering (3) and (4) synthetically, we ﬁnd that the results of granularity selection in IOMRIFS and IIMRIFS are inconsistent, so we need to further compute the comprehensive accuracies of IIMRIFS. CAFA IO ( E) = 0.7896, CAFA IO ( E) = 0.8125, CAFA IO ( E) = 0.7544; 2
CAF
A2IO
5
4
( E) = 0.8725, CAFA IO ( E) = 0.886, CAFA IO ( E) = 0.8588. 5
4
Through the above calculation results, we can see that the comprehensive score and comprehensive accuracy of granularity A4 are higher than A2 and A5 in the case of pessimistic multi granulation when the requirements are stricter. Therefore, the site 4 is eventually chosen as the optimal evaluation site.
228
Symmetry 2018, 10, 662
7. Conclusions In this paper, we propose two new granularity importance degree calculating methods among multiple granularities, and a granularity reduction algorithm is further developed. Subsequently, we design four novel MRIFS models based on reduction sets under optimistic and IMRS, i.e., OOMRIFS, OIMRIFS, IOMRIFS, and IIMRIFS, and further demonstrate their relevant properties. In addition, four threeway decisions models with novel MRIFS for the issue of internal redundant objects in reduction sets are constructed. Finally, we designe the comprehensive score function and the comprehensive precision function for the optimal granularity selection results. Meanwhile, the validity of the proposed models is veriﬁed by algorithms and examples. The works of this paper expand the application scopes of MRIFS and threeway decisions theory, which can solve issues such as spam email ﬁltering, risk decision, investment decisions, and so on. A question worth considering is how to extend the methods of this article to ﬁt the big data environment. Moreover, how to combine the fuzzy methods based on triangular or trapezoidal fuzzy numbers with the methods proposed in this paper is also a research problem. These issues will be investigated in our future work. Author Contributions: Z.A.X. and D.J.H. initiated the research and wrote the paper, M.J.L. participated in some of these search work, and M.Z. supervised the research work and provided helpful suggestions. Funding: This research received no external funding. Acknowledgments: This work is supported by the National Natural Science Foundation of China under Grant Nos. 61772176, 61402153, and the Scientiﬁc And Technological Project of Henan Province of China under Grant Nos. 182102210078, 182102210362, and the Plan for Scientiﬁc Innovation of Henan Province of China under Grant No. 18410051003, and the Key Scientiﬁc And Technological Project of Xinxiang City of China under Grant No. CXGG17002. Conﬂicts of Interest: The authors declare no conﬂicts of interest.
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SS symmetry Article
Maximum Detour–Harary Index for Some Graph Classes Wei Fang 1 , WeiHua Liu 2, *, JiaBao Liu 3 , FuYuan Chen 4 and ZhenMu Hong 5 and ZhengJiang Xia 5 1 2 3 4 5
*
College of Information & Network Engineering, Anhui Science and Technology University, Fengyang 233100, China; [email protected] College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, China School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China; [email protected] Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China; [email protected] School of Finance, Anhui University of Finance and Economics, Bengbu 233030, China; [email protected] (Z.M.H.); [email protected] (Z.J.X.) Correspondence: [email protected]
Received: 12 September 2018; Accepted: 22 October 2018; Published: 7 November 2018
Abstract: The deﬁnition of a Detour–Harary index is ωH ( G ) = 12 ∑u,v∈V (G) l (u,v1 G) , where G is a simple and connected graph, and l (u, v G ) is equal to the length of the longest path between vertices u and v. In this paper, we obtained the maximum Detour–Harary index about unicyclic graphs, bicyclic graphs, and cacti, respectively. Keywords: Detour–Harary index; maximum; unicyclic; bicyclic; cacti
1. Introduction In recent years, chemical graph theory (CGT) has been fastgrowing. It helps researchers to understand the structural properties of a molecular graph, for example, References [1–3]. A simple graph is an undirected graph without multiple edges and loops. Let G be a simple and connected graph, and V ( G ) and E( G ) be the vertex set and edge set of G, respectively. For vertices u, v of G, dG (v1 , v2 ) (or d(v1 , v2 ) for short) is the distance between v1 and v2 , which equals to the length of the shortest path between v1 and v2 in G; l (v1 , v2  G ) (or l (v1 , v2 ) for short) is the detour distance between v1 and v2 , which equals to the longest path of a shortest path between v1 and v2 in G. G [S] is an induced subgraph of G, the vertex set is S, and the edge set is the set of edges of G and both ends in S. G − S is the induced subgraph G [V ( G ) \ S]; when S = {w}, we write G − w for short. In 1947, Wiener introduced the ﬁrst molecular topological index–Wiener index. The Wiener index has applications in many ﬁelds, such as chemistry, communication, and cryptology [4–7]. Moreover, the Wiener index was studied from a purely graphtheoretical point of view [8–10]. In Reference [11], Wiener gave the deﬁnition of the Wiener index: W (G) =
1 d(u, v). 2 u,v∈∑ V (G)
The Harary index was independently introduced by Plavši´c et al. [12] and by Ivanciuc et al. [13] in 1993. In References [12,13], they gave the deﬁnition of the Harary index: H (G) =
Symmetry 2018, 10, 608; doi:10.3390/sym10110608
1 1 . 2 u,v∈∑ d ( u, v) V (G)
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Symmetry 2018, 10, 608
In Reference [13], Ivanciuc gave the deﬁnition of the Detour index: ω(G) =
1 l (u, v G ). 2 u,v∈∑ V (G)
Lukovits [14] investigated the use of the Detour index in quantitative structure–activity relationship (QSAR) studies. Trinajsti´c and his collaborators [15] analyzed the use of the Detour index, and compared its application with Wiener index. They found that the Detour index in combination with the Wiener index is very efﬁcient in the structureboiling point modeling of acyclic and cyclic saturated hydrocarbons. In this paper, we introduce a new graph invariant reciprocal to the Detour index, namely, the Detour–Harary index, as 1 1 . ωH ( G ) = 2 u,v∈∑ l ( u, v G ) V (G) Let G be a simple and connected graph, V ( G ) = n and E( G ) = m. If m = n − 1, then G is a tree; if m = n, then G is a unicyclic graph; if m = n + 1, then G is a bicyclic graph. Suppose Un (Bn , respectively) is the set of unicyclic (bicyclic, respectively) graphs set with n vertices. Any bicyclic graph G can be obtained from θ ( p, q, l )graph or θ ( p, q, l )graph G0 by attaching trees to the vertices, where p, q, l ≥ 1, and at most one of them is equal to 1. We denote G0 be the kernel of G (Figure 1). If each block of G is either a cycle or an edge, then we called graph G a cactus graph. Suppose Cnk be the set of all cacti with nvertices and k cycles. Obviously, Cn0 are trees, Cn1 are unicyclic graphs, and Cn2 are bicyclic graphs with exactly two cycles.
v
Pp+1 • •··· • • Pl+1 TT u • • • · · · • • T• v T Pq+1 TT • • · · · • • θ(p, q, l)
v
Cp • 1 •· · · • •l Cq
∞(p, q, l)
Figure 1. ∞graph and θgraph.
There are more results about cacti and bicyclic graphs [16–25]. More results about Harary index can be found in References [26–34], and more results about Detour index can be found in References [14,35–39]. Note that the Detour–Harary index is the same as Harary index for a tree graph; we study the Detour–Harary index of topological structures containing cycles. In this paper, we gave the maximum Detour–Harary index among Un ,Bn and Cnk (k ≥ 3), respectively. 2. Preliminaries In this section, we introduce useful lemmas and graph transformations. Lemma 1. [40] Let G be a connected graph, x be a cutvertex of G, and u and v be vertices occurring in different components that arise upon the deletion of vertex x. Then l (u, v G ) = l (u, x  G ) + l ( x, v G ).
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2.1. EdgeLifting Transformation The edgelifting transformation [41]. Let G1 and G2 be two graphs with n1 ≥ 2 and n2 ≥ 2 vertices. u0 ∈ V ( G1 ) and v0 ∈ V ( G2 ), G is the graph obtained from G1 and G2 by adding an edge between u0 and v0 . G is the graph obtained by identifying u0 to v0 and adding a pendent edge to u0 (v0 ). We called graph G the edgelifting transformation of graph G (see Figure 2).
w • 0 G1
u0 •
• v0
G2
G1
−−−−−−−−−−−−−−−−−→
u0 •
G2
Edgelifting transformation
G
G Figure 2. Edgelifting transformation.
Lemma 2. Let graph G be the edgelifting transformation of graph G. Then ωH ( G ) < ωH ( G ). Proof. By the deﬁnition of ωH ( G ) and Lemma 1, ωH ( G ) = ωH ( G1 ) + ωH ( G2 ) +
∑
x ∈V ( G1 )\{u0 }
1 + + l (u0 , v0  G ) x∈V (G∑)\{u
0} 1 y∈V ( G2 )\{v0 }
= ωH ( G1 ) + ωH ( G2 ) + +1+
∑
x ∈V ( G1 )\{u0 } y∈V ( G2 )\{v0 }
∑
x ∈V ( G1 )\{u0
1 + ∑ 1 + l ( u 0 , xG) } y∈V ( G )\{v 2
∑
x ∈V ( G1 )\{u0 }
0} 1 y ∈V ( G2 )\{u0 }
= ωH ( G1 ) + ωH ( G2 ) +
∑
0}
1 l ( x, y G )
1 + + ∑ l ( u 0 , w0  G ) x ∈ V ( G )\{u
x ∈V ( G1 )\{u0 } y ∈V ( G2 )\{u0 }
2
1 l ( u0 , y  G )
0
1 1 + l ( v 0 , y G ) }
1 , l ( u0 , x  G ) + 1 + l ( v0 , y  G )
ωH ( G ) = ωH ( G1 ) + ωH ( G2 ) +
+1+
1 + l (v0 , x  G ) y∈V (G∑)\{v
∑
1 + ∑ l ( w0 , x  G ) y ∈ V ( G )\{u 2
0}
1 l ( w0 , y  G )
1 l ( x , y  G )
x ∈V ( G1 )\{u0 }
1 + ∑ 1 + l ( u0 , x  G ) y ∈V ( G )\{u 2
0
1 1 + l ( u 0 , y G ) }
1 . l ( u0 , x  G ) + l ( u0 , y  G )
Obviously, ωH ( G1 ) = ωH ( G1 ); ωH ( G2 ) = ωH ( G2 );
l (u0 , x  G ) = l (u0 , x  G ), where x ∈ V ( G1 ) \ {u0 } and x ∈ V ( G1 ) \ {u0 }; l (v0 , y G ) = l (u0 , y  G ), where y ∈ V ( G2 ) \ {v0 } and y ∈ V ( G2 ) \ {u0 }.
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Then ωH ( G ) − ωH ( G ) =
∑
x ∈V ( G1 )\{u0 } y∈V ( G2 )\{v0 }
−
1 l ( x, u0  G ) + 1 + l (v0 , y G )
∑
x ∈V ( G1 )\{u0 } y ∈V ( G2 )\{u0 }
1 < 0. l ( x , u0  G ) + l ( u0 , y  G )
2.2. CycleEdge Transformation Suppose G ∈ Cnl is a cactus as shown in Figure 3. C p = v1 v2 · · · v p v1 is a cycle of G; Gi is a cactus, and vi ∈ V ( Gi ), 1 ≤ i ≤ p; Wvi = NG (vi ) ∩ V ( Gi ), 1 ≤ i ≤ p. G is the graph obtained from G by deleting the edges from vi to Wvi (2 ≤ i ≤ p), while adding the edges from v1 to Wvi (2 ≤ i ≤ p). We called graph G the cycleedge transformation of graph G (see Figure 3).
G1
G2 • v2 G3 • v3 ··
·
• v2 v1 • G1
−−−−−−−−−−−−−−−−−→ • v3 Cycle edge transformation ·· ·
vp • Gp
G2 − v2 ·· · Gp − vp
v1 • vp • G
G Figure 3. Cycleedge transformation.
Lemma 3. Suppose G ∈ Cnl is a cactus, p ≥ 3, and G is the cycleedge transformation of G (see Figure 3). Then, ωH ( G ) ≤ ωH ( G ), and the equality holds if and only if G ∼ = G . Proof. Let Vi = V ( Gi − vi ), 1 ≤ i ≤ p. By the deﬁnition of ωH ( G ) and Lemma 1,
ωH ( G ) = ωH (C p ) +
p
p
i
= ωH (C p ) +
i
y∈Vj i = j
p
p
p
∑
p
∑
x ∈Vi y ∈V ( C p )
1 l ( x, y G )
1 1 1 1 ∑ l (x, yG) + 2 ∑ ∑ ∑ l (x, vi G) + l (vi , v j G) + l (v j , yG) 2 i∑ =1 x,y∈V i =1 j =1 x ∈V i
+∑
p
p
1 1 1 1 ∑ l (x, yG) + 2 ∑ ∑ ∑ l (x, yG) + ∑ 2 i∑ =1 x,y∈V i =1 j =1 x ∈V i =1
i =1 x ∈Vi y ∈V ( C p )
i
y∈Vj i = j
1 , l ( x, vi  G ) + l (vi , y G )
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ωH ( G ) = ωH (C p ) +
p
p
p
i
= ωH (C p ) +
i
y∈Vj i = j
p
p
p
∑
x ∈Vi y ∈V ( C p )
1 l ( x, y G )
p
1 1 1 1 ∑ l (x, yG ) + 2 ∑ ∑ ∑ l (x, v1 G ) + l (v1 , yG ) 2 i∑ =1 x,y∈V i =1 j =1 x ∈V i
+∑
p
1 1 1 1 ∑ l (x, yG ) + 2 ∑ ∑ ∑ l (x, yG ) + ∑ 2 i∑ =1 x,y∈V i =1 j =1 x ∈V i =1
∑
i =1 x ∈Vi y ∈V ( C p )
i
y∈Vj i = j
1 . l ( x, v1  G ) + l (v1 , y G )
Obviously, p
∑ ∑
i =1 x,y∈Vi
p
1 = l ( x, y G )
∑ ∑
i =1 x,y∈Vi
1 ; l ( x, y G )
l ( x, vi  G ) = l ( x, v1  G ), where x ∈ Vi ; l (v j , y G ) = l (v1 , y G ), where y ∈ Vj ; p
∑ ∑
i =1 x ∈Vi y ∈V ( C p )
p
1 = l ( x, vi  G ) + l (vi , y G )
∑ ∑
i =1 x ∈Vi y ∈V ( C p )
1 . l ( x, v1  G ) + l (v1 , y G )
Then ωH ( G ) − ωH ( G ) =
p
p
1 1 ∑ ∑ l (x, vi G) + l (vi , v j G) + l (v j , yG) 2 i∑ =1 j =1 x ∈V i
y∈Vj i = j
−
p
p
1 1 ∑ ∑ l (x, v1 G ) + l (v1 , yG ) < 0. 2 i∑ =1 j =1 x ∈V i
y∈Vj i = j
The proof is completed. 2.3. Cycle Transformation Suppose G ∈ Cnl is a cactus, as shown in Figure 4. C p = v1 v2 · · · v p v1 is a cycle of G, and G1 is a simple and connected graph, v1 ∈ V ( G1 ). G is the graph obtained from G by deleting the edges from vi to vi+1 (2 ≤ i ≤ p − 1), meanwhile, adding the edges from v1 to vi (3 ≤ i ≤ p − 1). We called graph G is the cycle transformation of G (see Figure 4).
v2 •
• v2 v3 • ···
vp •
v1 • G 1
−−−−−−−−−−−−−→ Cycle transformation
@ @ [email protected]•
E
E vp • • •···•E
G1
v3 v4 vp−1 G
G Figure 4. Cycle transformation.
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Lemma 4. Suppose graph G is a simple and connected graph with p ≥ 4, and G is the cycle transformation of G(see Figure 4). Then, ωH ( G ) < ωH ( G ). Proof. Let V (C p ) = {v1 , v2 , · · · , v p }, V1 = V (C p − v1 ), V2 = V ( G1 − v1 ). By the deﬁnition of ωH ( G ), ωH ( G ) = ωH ( G1 ) +
= ωH ( G1 ) + ωH ( G ) = ωH ( G1 ) +
= ωH ( G1 ) +
∑
x,y∈V (C p )
∑
x,y∈V (C p )
∑
x,y∈V (C p )
∑
x,y∈V (C p )
1 1 + l ( x, y G ) x∑ l ( x, y G ) ∈V 1
y∈V2
1 1 + ∑ , l ( x, y G ) x∈ l ( x, v  G ) + l ( v1 , y  G ) 1 V, 1
y∈V2
1 1 + ∑ l ( x, y G ) x∈ l ( x, y G ) V, 1
y∈V2
1 1 + ∑ , l ( x, y G ) x∈ l ( x, v1  G ) + l (v1 , y G ) V, 1
y∈V2
Obviously, l ( x, y G ) ≥ l ( x, y G ), where x, y ∈ V1 ; l ( x, v1  G ) > 2 ≥ l ( x, v1  G ), where x ∈ V1 ; l (v1 , y G ) = l (v1 , y G ), where y ∈ V2 . Then ωH ( G ) − ωH ( G ) = (
∑
x,y∈V (C p )
1 − l ( x, y G ) x,y∈∑ V (C
p)
1 ) l ( x, y G )
1 1 − ∑ ) < 0. +( ∑ l ( x, v1  G ) + l (v1 , y G ) x∈ l ( x, v1  G ) + l (v1 , y G ) x ∈V , V, 1
1
y∈V2
y∈V2
3. Maximum Detour–Harary Index of Unicyclic Graphs For any unicyclic graph G ∈ Un , by repeating edgelifting transformations, cycleedge transformations, cycle transformations, or any combination of these on G, we get U1 from G, where graph U1 is deﬁned in Figure 5.
v2 •
@ v @ 1 4 @• ·· @·
n−3
v3 • U1 Figure 5. Unicyclic graph U1 .
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Symmetry 2018, 10, 608
Theorem 1. Let U1 be deﬁned as Figure 5. Then, U1 is the unique graph that attains the maximum 2 n −6 Detour–Harary index among all graphs in Un (n ≥ 3), and ωH (U1 ) = 3n − . 12 Proof. By Lemmas 2–4, U1 is the unique graph which attains the maximum Detour–Harary index of all graphs in Un . We then calculate the value ωH (U1 ). Let V (U1 ) = {v1 , v2 , · · · , vn }. It can be checked directly that n
1
∑ l (v1 , vi U1 )
i =2
∑
1≤i ≤n,i =2
= n − 2;
1 = l (v2 , vi U1 )
∑
1≤ j≤n,j =3
1 1 1 n−3 n = + + = ; l (v3 , v j U1 ) 2 2 3 3
1 n−4 2 3n − 2 = 1+ + = . l ( v , v  U ) 2 3 6 4 1 i 1≤i ≤n,i =4
∑
Then ωH (U1 ) =
=
n 1 n 1 1 1 +2 ∑ + ( n − 3) ∑ ] [ 2 i∑ l ( v , v  U ) l ( v , v  U ) l (v4 , vi U1 ) 2 i 1 i 1 1 =2 i =1 1≤i ≤n,i =2
3n2 − n − 6 . 12
The proof is completed. 4. Maximum Detour–Harary Index of Bicyclic Graphs For any bicyclic graph G ∈ ∞( p, q, l ) with exactly two cycles, by repeating edgelifting transformations, cycleedge transformations, cycle transformations, or any combination of these on G, we get B1 from G, where graph B1 is deﬁned in Figure 6. For any bicyclic graph G ∈ θ ( p, q, l ) with n vertices, by repeating edgelifting transformations on G, we get B2 from G, where graph B2 is deﬁned in Figure 7.
v2 •@
• v4
n−5 789: @ ··· @A [email protected] 1A• @
@ @
@• v5
v3 • B1
Figure 6. Bicyclic graph B1 .
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Symmetry 2018, 10, 608
··· [email protected]•
··· v1
@•
···
@•
k1 789: ··· A v1A•
uq
@
@ @
3 Hv 2 • k2 ··· H @
··@• vp−1 ·· uq−1 • ·· · · @· ·· ·· · · v v u • t+1 • t • t+1 · ·@· · ·@· · ·@·
u
3 4 @•·· l HH· 3
@
@
[email protected] 3•
A
· · ·A 9:78 k3
B2 (t ≥ 2)
B2 (p = q = 3, t = 2) Figure 7. Bicyclic graph B2 (t ≥ 2).
Theorem 2. Let B2 , B3 be deﬁned as Figures 7 and 8. Then, ωH ( B2 ) ≤ ωH ( B3 ), and the equality holds if and only if B2 ∼ = B3 .
n−4 789: ··· A v1A• @
@
@ u3 @•
v2 •
@
@ @
[email protected] 3•
Figure 8. Bicyclic graph B2 (t ≥ 2).
Proof. Case 1. B2 = B3 . Obviously, ωH ( B2 ) = ωH ( B3 ). Case 2. B2 = B3 and p = q = 3, t = 2(see Figures 7 and 8). Let V1 = {v1 , v2 , v3 , u3 }, Wvi = {w  wvi ∈ E( B2 ) and d B2 (w) = 1} and Wvi  = k i , Wu3 = {w  wu3 ∈ E( B2 ) and d B2 (w) = 1} and Wu3  = l3 , k i + l3 = n − 4 for 1 ≤ i ≤ 3. ωH ( B2 ) =
ωH ( B3 ) =
∑
x,y∈V1
∑
x,y∈V1
1 + l ( x, y B2 ) 1 + l ( x, y B3 )
∑
x ∈V1 , y∈V ( B2 )−V1
∑
x ∈V1 , y∈V ( B3 )−V1
1 1 + , l ( x, y B2 ) x,y∈V∑ l ( x, y  B2 ) ( B )−V 2
1
1 1 + . l ( x, y B3 ) x,y∈V∑ l ( x, y  B3 ) ( B )−V 3
1
Easily,
∑
x,y∈V1
1 = l ( x, y B2 )
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∑
x,y∈V1
1 l ( x, y B3 )
(1)
Symmetry 2018, 10, 608
∑
x ∈V1 , y∈V ( B2 )−V1
1 = l ( x, y B2 )
1 1 + l (v1 , w B2 ) w∈V (∑ l ( v , w B2 ) 2 B )−V
∑
w∈V ( B2 )−V1
2
∑
+
w∈V ( B2 )−V1
1
1 1 + l (v3 , w B2 ) w∈V (∑ l ( u , w B2 ) 3 B )−V 2
1
1 1 1 1 1 1 = ( 1 · k 1 + · k 2 + · k 3 + · l3 ) + ( · k 1 + 1 · k 2 + · k 3 + · l3 ) 4 3 4 4 4 4 1 1 1 1 1 1 + ( · k 1 + · k 2 + 1 · k 3 + · l3 ) + ( · k 1 + · k 2 + · k 3 + 1 · l3 ) 3 4 4 4 4 4 11(k1 + k3 ) 7(k2 + l3 ) + = , 6 4
∑
x ∈V1 , y∈V ( B3 )−V1
1 = l ( x, y B3 )
∑
w∈V ( B3 )−V1
+
∑
1 1 + l (v1 , w B3 ) w∈V (∑ l (v2 , w B3 ) B )−V 3
w∈V ( B3 )−V1
1
1 1 + l (v3 , w B3 ) w∈V (∑ l (u3 , w B3 ) B )−V 3
1
1 1 1 = 1 · ( n − 4) + · ( n − 4) + · ( n − 4) + · ( n − 4) 4 3 4 11(n − 4) = 6 11(k1 + k2 + k3 + l3 ) , = (since k i + l3 = n − 4 for 1 ≤ i ≤ 3) 6 Then,
1 − l ( x, y B2 )
∑
x ∈V1 , y∈V ( B2 )−V1
∑
x ∈V1 , y∈V ( B3 )−V1
1 1 = (k2 + l3 ) ≥ 0, l ( x, y B3 ) 12
(2)
the equality holds if and only if k2 = l3 = 0. On the other hand l ( x,y1 B ) ≤ l ( x,y1 B ) = 12 , where x, y ∈ V ( B2 ) − V1 , then 2
3
∑
x,y∈V ( B3 )−V1
1 ≤ l ( x, y B2 )
∑
x,y∈V ( B3 )−V1
1 , l ( x, y B3 )
(3)
the equality holds if k1 = n − 4 or k2 = n − 4 or k3 = n − 4 or l3 = n − 4. By (1)–(3) and B2 = B3 , we have ωH ( B2 ) < ωH ( B3 ). Case 3. B2 = B3 and p + q − t > 4. It can be checked directly that ; < ωH ( B2 ) ≤ (1 + 1 + · · · + 1) + 12 (n− p2−q+t) + 14 (n2 ) − (n − p − q + t) − (n− p2−q+t) , 9 :7 8 n−p−q+t 4 1 1 ωH ( B3 ) = (1 + 1 + · · · + 1) + 12 [1 + (n− 2 )] + 3 [5 + ( n − 4)] + 4 [2( n − 4)]. 9 :7 8 n−4 B2 , B3 are bicyclic graphs and  V ( B2 ) = V ( B3 ) = n. Since p + q − t > 4, then n − p − q + t ≤ 4 n − 5 and (n− p2−q+t) < (n− 2 ), we have ωH ( B2 ) < ωH ( B3 ). The proof is completed. Theorem 3. Let B1 , B3 be deﬁned as Figures 6 and 8. Then, ( max{ωH (Bn )} =
ωH ( B3 ) =
13 6 ,
ωH ( B1 ) = ωH ( B3 ) = 240
if n = 4, 3n2 −5n−2 , 12
if n ≥ 5.
Symmetry 2018, 10, 608
Proof. Let G ∈ ∞( p, q, l ), by Lemmas 2–4, we have ωH ( G ) ≤ ωH ( B1 ), and the equality holds if and only if G ∼ = B1 . For any bicyclic graph with G ∈ θ ( p, q, l ), by Lemmas 2–4 and Theorem 2, we have ωH ( G ) ≤ ωH ( B3 ), and the equality holds if and only if G ∼ = B3 . Thus, max{ωH (Bn )} = max{ωH ( B1 ), ωH ( B3 )}. It can be checked directly that 1 n−5 1 1 3n2 − 5n − 2 ωH ( B1 ) = (n − 5) + [ + 6] + [4(n − 5)] + · 4 = , n ≥ 5; 2 3 4 12 2 ωH ( B3 ) = (n − 4) +
1 n−4 1 1 3n2 − 5n − 2 + (n − 4) + [2(n − 4)] = , n ≥ 4. 2 3 4 12 2
Therefore ( max{ωH (Bn )} =
ωH ( B3 ) =
if n = 4,
13 6 ,
ωH ( B1 ) = ωH ( B3 ) =
3n2 −5n−2 , 12
if n ≥ 5.
The proof is completed. 5. Maximum Detour–Harary Index of Cacti For any cactus graph G ∈ Cnk (k ≥ 3), by repeating edgelifting transformations, cycleedge transformations, cycle transformations, or any combination of these on G, we get C1 from G, where graph C1 is deﬁned in Figure 9.
v2k • v2k+1•
· · ·
•
v5 •
v1 •
v4 •
v3
• v2
• ··· • v2k+2 vn Figure 9. Cactus graph C1 (k ≥ 3).
Theorem 4. Let C1 be deﬁned as Figure 9. Then, C1 is the unique cactus graph in Cnk (k ≥ 3) that attains the 2 2 +3n−2k−6 . maximum Detour–Harary index, and ωH (C1 ) = 3n +2k −4nk 12 Proof. By Lemmas 2–4, C1 is the unique graph that attains the maximum Detour–Harary index of all graphs in Cnk (k ≥ 3).
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Symmetry 2018, 10, 608
Let V (C1 ) = {v1 , v2 , · · · , vn }, and it can be checked directly that n
1
∑ l (v1 , vi C1 )
i =2
= 1 · (n − 2k − 1) +
1 · 2k = n − k − 1; 2
1 1 1 1 1 1 1 = · 2 + · (n − 2k − 1) + · (2k − 2) = n − k + ; l ( v , v C ) 2 3 4 3 6 6 2 i 1 1≤i ≤n,i =2
∑
n −1
∑
j =1
1 1 1 1 1 = 1 + · (n − 2k − 2) + · 2k = n − k. l (vn , v j C1 ) 2 3 2 3
Then, 1 1 1 1 1 1 [(n − k − 1) + 2k · ( n − k + ) + (n − 2k − 1) · ( n − k)] 2 3 6 6 2 3 3n2 + 2k2 − 4nk + 3n − 2k − 6 . = 12
ωH (C1 ) =
The proof is completed. Author Contributions: Conceptualization, W.F. and W.H.L.; methodology, F.Y.C.; Z.J.X. and J.B.L.; writing—original draft preparation, W.F. and Z.M.H; writing—review and editing, W.H.L. Funding: This research was funded by NSFC Grant (No.11601001, No.11601002, No.11601006). Conﬂicts of Interest: The authors declare no conﬂict of interest.
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Chen, S. Cacti with the smallest, second smallest, and third smallest Gutman index. J. Comb. Optim. 2016, 31, 327–332. [CrossRef] Chen, Z.; Dehmer, M.; Shi, Y.; Yang, H. Sharp upper bounds for the Balaban index of bicyclic graphs. MATCH Commun. Math. Comput. Chem. 2016, 75, 105–128. Fang, W.; Gao, Y.; Shao, Y.; Gao, W.; Jing, G.; Li, Z. Maximum Balaban index and sumBalaban index of bicyclic graphs. MATCH Commun. Math. Comput. Chem. 2016, 75, 129–156. Fang, W.; Wang, Y.; Liu, J.B.; Jing, G. Maximum ResistanceHarary index of cacti. Discret. Appl. Math. 2018. [CrossRef] Gutman, I.; Li, S.; Wei, W. Cacti with nvertices and tcycles having extremal Wiener index. Discret. Appl. Math. 2017, 232, 189–200. [CrossRef] Ji, S.; Li, X.; Shi, Y. Extremal matching energy of bicyclic graphs. MATCH Commun. Math. Comput. Chem. 2013, 70, 697–706. Liu, J.; Pan, X.; Yu, L.; Li, D. Complete characterization of bicyclic graphs with minimal Kirchhoff index. Discret. Appl. Math. 2016, 200, 95–107. [CrossRef] Wang, H.; Hua, H.; Wang, D. Cacti with minimum, secondminimum, and thirdminimum Kirchhoff indices. Math. Commun. 2010, 15, 347–358. Wang, L.; Fan, Y.; Wang, Y. Maximum Estrada index of bicyclic graphs. Discret. Appl. Math. 2015, 180, 194–199. [CrossRef] Lu, Y.; Wang, L.; Xiao, P. Complex Unit Gain Bicyclic Graphs with Rank 2,3 or 4. Linear Algebra Appl. 2017, 523, 169–186. [CrossRef] Furtula, B.; Gutman, I.; Katani´c, V. Threecenter Harary index and its applications. Iran. J. Math. Chem. 2016, 7, 61–68. Feng, L.; Lan, Y.; Liu, W.; Wang, X. Minimal Harary index of graphs with small parameters. MATCH Commun. Math. Comput. Chem. 2016, 76, 23–42. Hua, H.; Ning, B. Wiener index, Harary index and hamiltonicity of graphs. MATCH Commun. Math. Comput. Chem. 2017, 78, 153–162. Li, X.; Fan, Y. The connectivity and the Harary index of a graph. Discret. Appl. Math. 2015, 181, 167–173. [CrossRef] Xu, K.; Das, K.C. On Harary index of graphs. Discret. Appl. Math. 2011, 159, 1631–1640. [CrossRef] Xu, K. Trees with the seven smallest and eight greatest Harary indices. Discret. Appl. Math. 2012, 160, 321–331. [CrossRef] Xu, K.; Das, K.C. Extremal unicyclic and bicyclic graphs with respect to Harary Index. Bull. Malaysian Math. Sci. Soc. 2013, 36, 373–383. Xu, K.; Wang, J.; Das, K.C.; Klavžar, S. Weighted Harary indices of apex trees and kapex trees. Discret. Appl. Math. 2015, 189, 30–40. [CrossRef] Zhou, B.; Cai, X.; Trinajsti´c, N. On Harary index. J. Math. Chem. 2008, 44, 611–618. [CrossRef] Fang, W.; Yu, H.; Gao, Y.; Jing, G.; Li, Z.; Li, X. Minimum Detour index of cactus graphs. Ars Comb. 2019, in press. Qi, X.; Zhou, B. Detour index of a class of unicyclic graphs. Filomat 2010, 24, 29–40. Qi, X.; Zhou, B. HyperDetour index of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 2011, 66, 329–342. Rücker, G.; Rücker, C. Symmetryaided computation of the Detour matrix and the Detour index. J. Chem. Inf. Comput. Sci. 1998, 38, 710–714. [CrossRef] Zhou, B.; Cai, X. On Detour index. MATCH Commun. Math. Comput. Chem. 2010, 44, 199–210. Qi, X. Detour index of bicyclic graphs. Util. Math. 2013, 90, 101–113. Deng, H. On the Balaban index of trees. MATCH Commun. Math. Comput. Chem. 2011, 66, 253–260. c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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QFilters of Quantum BAlgebras and Basic Implication Algebras Xiaohong Zhang 1,2, *, Rajab Ali Borzooei 3 and Young Bae Jun 3,4 1 2 3 4
*
Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran; [email protected] (R.A.B.); [email protected] (Y.B.J.) Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea Correspondence: [email protected] or [email protected]
Received: 26 September 2018; Accepted: 26 October 2018; Published: 1 November 2018
Abstract: The concept of quantum Balgebra was introduced by Rump and Yang, that is, uniﬁed algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of qﬁlter in quantum Balgebra is proposed, and quotient structures are constructed by qﬁlters (in contrast, although the notion of ﬁlter in quantum Balgebra has been deﬁned before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a uniﬁed frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum Balgebra is not applied to nonassociative fuzzy logics). The ﬁlter theory of basic implication algebras is also established. Keywords: fuzzy implication; quantum Balgebra; qﬁlter; quotient algebra; basic implication algebra
1. Introduction For classical logic and nonclassical logics (multivalued logic, quantum logic, tnormbased fuzzy logic [1–6]), logical implication operators play an important role. In the study of fuzzy logics, fuzzy implications are also the focus of research, and a large number of literatures involve this topic [7–16]. Moreover, some algebraic systems focusing on implication operators are also hot topics. Especially with the indepth study of noncommutative fuzzy logics in recent years, some related implication algebraic systems have attracted the attention of scholars, such as pseudobasiclogic (BL) algebras, pseudo monoidal tnormbased logic (MTL) algebras, and pseudo B, C, K axiom (BCK)/ B, C, I axiom (BCI) algebras [17–23] (see also References [5–7]). For formalizing the implication fragment of the logic of quantales, Rump and Yang proposed the notion of quantum Balgebras [24,25], which provide a uniﬁed semantic for a wide class of nonclassical logics. Speciﬁcally, quantum Balgebras encompass many implication algebras, like pseudoBCK/BCI algebras, (commutative and noncommutative) residuated lattices, pseudo MV/BL/MTL algebras, and generalized pseudoeffect algebras. New research articles on quantum Balgebras can be found in References [26–28]. Note that all hoops and pseudohoops are special quantum Balgebras, and they are closely related to Lalgebras [29]. Although the deﬁnition of a ﬁlter in a quantum Balgebra is given in Reference [30], quotient algebraic structures are not established by using ﬁlters. In fact, ﬁlters in special subclasses of quantum Balgebras are mainly discussed in Reference [30], and these subclasses require a unital element. In this paper, by introducing the concept of a qﬁlter in quantum Balgebras, we establish the quotient structures using qﬁlters in a natural way. At the same time, although quantum Balgebra has generality, it cannot include the implication structure of nonassociative fuzzy logics [31,32], so we propose a wider Symmetry 2018, 10, 573; doi:10.3390/sym10110573
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concept, that is, basic implication algebra that can include a wider range of implication operations, establish ﬁlter theory, and obtain quotient algebra. 2. Preliminaries Deﬁnition 1. Let (X, ≤) be partially ordered set endowed with two binary operations → and [24,25]. Then, (X, →, , ≤) is called a quantum Balgebra if it satisﬁes: ∀x, y, z∈X, (1) (2) (3) (4)
y→z ≤ (x→y)→(x→z); yz ≤ (xy) (xz); y ≤ z ⇒ x→y ≤ x→z; x ≤ y→z ⇐⇒ y ≤ xz.
If u∈X exists, such that u→x = ux = x for all x in X, then u is called a unit element of X. Obviously, the unit element is unique. When a unit element exists in X, we call X unital. Proposition 1. An algebra structure (X, →, , ≤) endowed with a partially order ≤ and two binary operations → and is a quantum Balgebra if and only if it satisﬁes [4]: ∀x, y, z∈X, (1) (2) (3)
x→(yz) = y(x→z); y ≤ z ⇒ x→y ≤ x→z; x ≤ y→z ⇐⇒ y ≤ xz.
Proposition 2. Let (X, →, , ≤) be a quantum Balgebra [24–26]. Then, (∀ x, y, z∈X) y ≤ z ⇒ xy ≤ xz; y ≤ z ⇒ zx ≤ yx; y ≤ z ⇒ z→x ≤ y→x; x ≤ (xy)→y, x ≤ (x→y)y; x→y = ((x→y)y)→y, xy = ((xy)→y)y; x→y ≤ (y→z)(x→z); xy ≤ (yz)→(xz); assume that u is the unit of X, then u ≤ xy ⇐⇒ x ≤ y ⇐⇒ u ≤ x→y; if 0∈X exists, such that 0 ≤ x for all x in X, then 0 = 00 = 0→0 is the greatest element (denote by 1), and x→1 = x1=1 for all x∈X; (10) if X is a lattice, then (x ∨ y)→z = (x→z)∧(y→z), (x∨y)z = (xz) ∨(yz). (1) (2) (3) (4) (5) (6) (7) (8) (9)
Deﬁnition 2. Let (X, ≤) be partially ordered set and Y⊆ X [24]. If x ≥ y∈Y implies x∈Y, then Y is called to be an upper set of X. The smallest upper set containing a given x∈X is denoted by ↑x. For quantum Balgebra X, the set of upper sets is denoted by U(X). For A, B∈U(X), deﬁne A·B = {x∈X ∃b∈B: b→x∈A}. We can verify that A·B = {x∈X∃a∈A: ax∈B} = {x∈X∃a∈A, b∈B: a≤b→x} = {x∈X∃a∈A, b∈B: b ≤ ax}. Deﬁnition 3. Let A be an empty set, ≤ be a binary relation on A [17,18], → and be binary operations on A, and 1 be an element of A. Then, structure (A, →, , ≤, 1) is called a pseudoBCI algebra if it satisﬁes the following axioms: ∀ x, y, z∈A, (1) (2)
x→y ≤ (y→z)(x→z), xy ≤ (yz)→(xz), x ≤ (xy)→y, x ≤ (x→y)y; 245
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(3) (4) (5)
x ≤ x; x ≤ y, y ≤ x ⇒ x = y; x ≤ y ⇐⇒ x→y = 1 ⇐⇒ xy = 1.
If pseudoBCI algebra A satisﬁes: x→1 = 1 (or x1 = 1) for all x∈A, then A is called a pseudoBCK algebra. Proposition 3. Let (A, →, , ≤, 1) be a pseudoBCI algebra [18–20]. We have (∀x, y, z∈A) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
if 1 ≤ x, then x = 1; if x ≤ y, then y→z ≤ x→z and yz ≤ xz; if x ≤ y and y ≤ z, then x ≤ z; x→(yz) = y(x→z); x ≤ y→z ⇐⇒ y ≤ xz; x→y ≤ (z→x)→(z→y); xy ≤ (zx)(zy); if x ≤ y, then z→x ≤ z→y and zx ≤ zy; 1→x = 1x =x; y→x = ((y→x)x)→x, yx = ((yx)→x) x; x→y ≤ (y→x) 1, xy ≤ (yx)→1; (x→y)→1 = (x→1)(y→1), (xy)1 = (x1)→(y1); x→1 = x1.
Proposition 4. Let (A, →, , ≤, 1) be a pseudoBCK algebra [17], then (∀x, y∈A): x ≤ y→x, x ≤ yx. Deﬁnition 4. Let X be a unital quantum Balgebra [24]. If there exists x∈X, such that x→u = xu = u, then we call that x integral. The subset of integral element in X is denoted by I(X). Proposition 5. Let X be a quantum Balgebra [24]. Then, the following assertions are equivalent: (1) (2) (3)
X is a pseudoBCK algebra; X is unital, and every element of X is integral; X has the greatest element, which is a unit element.
Proposition 6. Every pseudoBCI algebra is a unital quantum Balgebra [25]. And, a quantum Balgebra is a pseudoBCI algebra if and only if its unit element u is maximal. Deﬁnition 5. Let (A, →, , ≤, 1) be a pseudoBCI algebra [20,21]. When the following identities are satisﬁed, we call X an antigrouped pseudoBCI algebra:
∀x∈A, (x→1)→1 = x or (x1)1 = x. Proposition 7. Let (A, →, , ≤, 1) be a pseudoBCI algebra [20]. Then, A is antigrouped if and only if the following conditions are satisﬁed: (G1) for all x, y, z∈A, (x→y)→(x→z) = y→z, and (G2) for all x, y, z∈A, (xy)(xz) = yz. Deﬁnition 6. Let (A, →, , ≤, 1) be a pseudoBCI algebra and F ⊆ X [19,20]. When the following conditions are satisﬁed, we call F a pseudoBCI ﬁlter (brieﬂy, ﬁlter) of X: (F1) 1∈F; 246
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(F2) x∈F, x→y∈F =⇒ y∈F; (F3) x∈F, xy∈F =⇒ y∈F. Deﬁnition 7. Let (A, →, , ≤, 1) be a pseudoBCI algebra and F be a ﬁlter of X [20,21]. When the following condition is satisﬁed, we call F an antigrouped ﬁlter of X: (GF) ∀x∈X, (x→1)→1∈F or (x1)1∈F =⇒ x∈F. Deﬁnition 8. A subset F of pseudoBCI algebra X is called a pﬁlter of X if the following conditions are satisﬁed [20,21]: (P1) 1∈F, (P2) (x→y)(x→z)∈F and y∈F imply z∈F, (P3) (xy)→(xz)∈F and y∈F imply z∈F. 3. QFilters in Quantum BAlgebra In Reference [30], the notion of ﬁlter in quantum Balgebra is proposed. If X is a quantum Balgebra and F is a nonempty set of X, then F is called the ﬁlter of X if F∈U(X) and F·F ⊆ F. That is, F is a ﬁlter of X, if and only if: (1) F is a nonempty upper subset of X; (2) (z∈X, y∈F, y→z∈F) ⇒ z∈F. We denote the set of all ﬁlters of X by F(X). In this section, we discuss a new concept of qﬁlter in quantum Balgebra; by using qﬁlters, we construct the quotient algebras. Deﬁnition 9. A nonempty subset F of quantum Balgebra X is called a qﬁlter of X if it satisﬁes: (1) (2) (3) (4)
F is an upper set of X, that is, F∈U(X); for all x∈F, x→x∈F and xx∈F; x∈F, y∈X, x→y∈F =⇒ y∈F. A qﬁlter of X is normal if x→y∈F⇐⇒ xy∈F.
Proposition 8. Let F be a qﬁlter of quantum Balgebra X. Then, (1) (2) (3)
x∈F, y∈X, xy∈F =⇒ y∈F. x∈F and y∈X =⇒ (xy)→y∈F and (x→y)y∈F. if X is unital, then Condition (2) in Deﬁnition 9 can be replaced by u∈F, where u is the unit element of X.
Proof. (1) Assume that x∈F, y∈X, and xy∈F. Then, by Proposition 2 (4), x ≤ (xy)→y. Applying Deﬁnition 9 (1) and (3), we get that y∈F. (2) Using Proposition 2 (4) and Deﬁnition 9 (1), we can get (2). (3) If X is unital with unit u, then u→u = u. Moreover, applying Proposition 2 (8), u ≤ xx and u ≤ x→x from x ≤ x, for all x∈X. Therefore, for unital quantum Balgebra X, Condition (2) in Deﬁnition 8 can be replaced by condition “u∈F”. By Deﬁnition 6, and Propositions 6 and 8, we get the following result (the proof is omitted). Proposition 9. Let (A, →, , ≤, 1) be a pseudoBCI algebra. Then, an empty subset of A is a qﬁlter of A (as a quantum Balgebra) if and only if it is a ﬁlter of A (according to Deﬁnition 6). Example 1. Let X = {a, b, c, d, e, f}. Deﬁne operations → and on X as per the following Cayley Tables 1 and 2; the order on X is deﬁned as follows: b ≤ a ≤ f; e ≤ d ≤ c. Then, X is a quantum Balgebra (we can verify
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it with the Matlab software (The MathWorks Inc., Natick, MA, USA)), but it is not a pseudoBCI algebra. Let F1 = {f}, F2 = {a, b, f}; then, F1 is a ﬁlter but not a qﬁlter of X, and F2 is a normal qﬁlter of X. Table 1. Cayley table of operation →.
→
a
b
c
d
e
f
a b c d e f
f f c c c a
a a c c c b
c c f f f c
c c a f f d
c c b a a e
f f c c c f
Table 2. Cayley table of operation .
a
b
c
d
e
f
a b c d e f
f f c c c a
a f c c c b
c c f f f c
c c a a f d
d c a a a e
f f c c c f
Theorem 1. Let X be a quantum Balgebra and F a normal qﬁlter of X. Deﬁne the binary ≈F on X as follows: x ≈F y ⇐⇒ x→y∈F and y→x∈F, where x, y∈X. Then, (1) (2)
≈F is an equivalent relation on X; ≈F is a congruence relation on X, that is, x≈F y =⇒ (z→x) ≈F (z→y), (x→z) ≈F (y→z), (zx) ≈F (zy), (xz) ≈F (yz), for all z∈X.
Proof. (1) For any x∈X, by Deﬁnition 9 (2), x→x∈F, it follows that x ≈F x. For all x, y∈X, if x ≈F y, we can easily verify that y ≈F x. Assume that x ≈F y, y ≈F z. Then, x→y∈F, y→x∈F, y→z∈F, and z→y∈F, since y→z ≤ (x→y)→(x→z) by Deﬁnition 1 (1). From this and Deﬁnition 9, we have x→z∈F. Similarly, we can get z→x∈F. Thus, x ≈F z. Therefore, ≈F is an equivalent relation on X. (2) If x ≈F y, then x→y∈F, y→x∈F. Since x→y ≤ (z→x)→(z→y), by Deﬁnition 1 (1). y→x ≤ (z→y)→(z→x), by Deﬁnition 1 (1). Using Deﬁnition 9 (1), (z→x)→(z→y)∈F, (z→y)→(z→x)∈F. It follows that (z→x) ≈F (z→y). Moreover, since x→y ≤ (y→z)(x→z), by Proposition 2 (6). y→x ≤ (x→z)(y→z), by Proposition 2 (6).
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Then, form x→y∈F and y→x∈F; using Deﬁnition 9 (1), we have (y→z)(x→z)∈F, (x→z)(y→z)∈F. Since F is normal, by Deﬁnition 9 we get (y→z)→(x→z)∈F, (x→z)→(y→z)∈F. Thus, (x→z) ≈F (y→z). Similarly, we can get that x ≈F y =⇒ (zx) ≈F (zy) and (xz) ≈F (yz). Deﬁnition 10. A quantum Balgebra X is considered to be perfect, if it satisﬁes: (1) (1)
for any normal qﬁlter F of X, x, y in X, (there exists an∈X, such that [x→y]F = [a→a]F ) ⇐⇒ (there exists b∈X, such that [xy]F = [bb]F ). for any normal qﬁlter F of X, (X/≈F →, , ≤) is a quantum Balgebra, where quotient operations → and are deﬁned in a canonical way, and ≤ is deﬁned as follows: [x]F ≤ [y]F ⇐⇒ (there exists a∈X such that [x]F →[y]F = [a→a]F ) ⇐⇒ (there exists b∈X such that [x]F [y]F = [bb]F ).
Theorem 2. Let (A, →, , ≤, 1) be a pseudoBCI algebra, then A is a perfect quantum Balgebra. Proof. By Proposition 6, we know that A is a quantum Balgebra. (1) For any normal qﬁlter F of A, x, y∈A, if there exists a∈A, such that [x→y]F = [a→a]F , then [x→y]F = [a→a]F = [1]F . It follows that (x→y)→1∈F, 1→(x→y) = x→y∈F. Applying Proposition 3 (11) and (12), we have (x→1)(y→1) = (x→y)→1∈F. Since F is normal, from (x→1)(y→1)∈F and x→y∈F we get that (x→1)→(y→1)∈F and xy∈F. Applying Proposition 3 (11) and (12) again, (xy)→1 = (x→1)→(y→1). Thus, (xy)→1 = (x→1)→(y→1)∈F and 1→(xy) = xy∈F. This means that [xy]F = [1]F = [11]F . Similarly, we can prove that the inverse is true. That is, Deﬁnition 10 (1) holds for A. (2) For any normal qﬁlter F of pseudoBCI algebra A, binary ≤ on A/ ≈F is deﬁned as the following: [x]F ≤ [y]F ⇐⇒ [x]F →[y]F = [1]F . We verify that ≤ is a partial binary on A/ ≈F . Obviously, [x]F ≤ [x]F for any x∈A. If [x]F ≤ [y]F and [y]F ≤ [x]F , then [x]F →[y]F = [x→y]F = [1]F , [y]F →[x]F = [y→x]F = [1]F . By the deﬁnition of equivalent class, x→y = 1→(x→y) ∈F, y→x = 1→(y→x)∈F. It follows that x ≈F y; thus, [x]F = [y]F . If [x]F ≤ [y]F and [y]F ≤ [z]F , then [x]F →[y]F = [x→y]F = [1]F , [y]F →[z]F = [y→z]F = [1]F . Thus, x→y = 1→(x→y)∈F, (x→y)→1∈F; y→z = 1→(y→z)∈F, (y→z)→1∈F. Applying Deﬁnition 3 and Proposition 3, y→z ≤ (x→y)→(x→z), 249
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(x→y)→1 = (x→1)(y→1) ≤ ([(y→1)(z→1)]→[(x→1)(z→1)]) = [(y→z)→1]→[(x→z)→1]. By Deﬁnition 9, 1→(x→z) = x→z∈F, (x→z)→1∈F. This means that (x→z) ≈F 1, [x→z]F = [1]F . That is, [x]F →[z]F =[x→z]F = [1]F , [x]F ≤ [z]F . Therefore, applying Theorem 1, we know that (A/≈F →, , [1]F ) is a quantum Balgebra and pseudoBCI algebra. That is, Deﬁnition 10 (2) holds for A. Hence, we know that A is a perfect quantum Balgebra. The following examples show that there are some perfect quantum Balgebras that may not be a pseudoBCI algebra. Example 2. Let X = {a, b, c, d, e, 1}. Deﬁne operations → and on X as per the following Cayley Tables 3 and 4, the order on X is deﬁned as the following: b ≤ a ≤ 1; e ≤ d ≤ c. Then, X is a pseudoBCI algebra (we can verify it with Matlab). Denote F1 = {1}, F2 = {a, b, 1}, F3 = X, then Fi (i = 1, 2, 3) are all normal qﬁlters of X, and quotient algebras (X/≈Fi →, , [1]Fi ) are pseudoBCI algebras. Thus, X is a perfect quantum Balgebra. Table 3. Cayley table of operation →.
→
a
b
c
d
e
1
a b c d e 1
1 1 c c c a
a 1 c c c b
c c 1 1 1 c
c c a 1 1 d
c c b a 1 e
1 1 c c c 1
Table 4. Cayley table of operation .
a
b
c
d
e
1
a b c d e 1
1 1 c c c a
a 1 c c c b
c c 1 1 1 c
c c a 1 1 d
d c a a 1 e
1 1 c c c 1
Example 3. Let X = {a, b, c, d, e, f}. Deﬁne operations → and on X as per the following Cayley Tables 5 and 6, the order on X is deﬁned as follows: b ≤ a ≤ f; e ≤ d ≤ c. Then, X is a quantum Balgebra (we can verify it with Matlab), but it is not a pseudoBCI algebra, since ee = e→e. Denote F = {a, b, f}, then F, X are all normal qﬁlters of X, quotient algebras (X/≈F →, , ≤), (X/≈X →, , ≤) are quantum Balgebras, and X is a perfect quantum Balgebra. Table 5. Cayley table of operation →.
→
a
b
c
d
e
f
a b c d e f
f f c c c a
a f c c c b
c c f f f c
c c a f f d
c c b a f e
f f c c c f
250
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Table 6. Cayley table of operation .
a
b
c
d
e
f
a b c d e f
f f c c c a
a f c c c b
c c f f f c
c c a f f d
d c a a a e
f f c c c f
4. Basic Implication Algebras and Filters =
Deﬁnition 11. Let (A, ∨, ∧, , →, 0, 1) be a type(2, 2, 2, 2, 0, 0) algebra [32]. A is called a nonassociative residuated lattice, if it satisﬁes: (A1) (A, ∨, ∧, 0, 1) is a bounded lattice; = (A2) (A, , 1) is a commutative groupoid with unit element 1; = (A3) ∀x, y, z∈A, x y ≤ z ⇐⇒ x ≤ y→z. Proposition 10. Let (A, ∨, ∧, (1) (2) (3) (4) (5) (6) (7) (8) (9)
=
, →, 0, 1) be a nonassociative residuated lattice [32]. Then, (∀ x, y, z∈A)
x ≤ y ⇐⇒ x→y = 1; = = x ≤ y ⇒ x z ≤ y z; x ≤ y ⇒ y→z ≤ x→z; x ≤ y ⇒ z→x ≤ z→y; = = = x (y∨z) = (x y)∨(x z); x→(y∧z) = (x→y)∧(x→z); (y∨z)→x = (y→x)∧(z→x); = (x→y) x ≤ x, y; (x→y)→y ≥ x, y.
Example 4. Let A = [0, 1], operation x Then, follows:
=
>
=
on A is deﬁned as follows:
y = 0.5xy + 0.5max{0, x + y − 1}, x, y∈A.
is a nonassociative tnorm on A (see Example 1 in Reference [32]). Operation → is deﬁned as x→y = max{z∈[0, 1]z
>
x ≤ y}, x, y∈A.
=
Then, (A, max, min, , →, 0, 1) is a nonassoiative residuated lattice (see Theorem 5 in Reference [32]). Assume that x = 0.55, y = 0.2, z = 0.1, then y → z = 0.2 → 0.1 = max{ a ∈ [0, 1 a
>
>
0.2 ≤ 0.1} =
5 . 6
17 . 31 > 4 0.55 ≤ 0.1} = x → z = 0.55 → 0.1 = max{ a ∈ [0, 1 a . 11 > 17 17 4 4 67 ( x → y) → ( x → z) = → = max{ a ∈ [0, 1 a ≤ }= . 31 11 31 11 88 x → y = 0.55 → 0.2 = max{ a ∈ [0, 1 a
0.55 ≤ 0.2} =
Therefore, y → z ( x → y ) → ( x → z ). 251
Symmetry 2018, 10, 573
Example 4 shows that Condition (1) in Deﬁnition 1 is not true for general nonassociative residuated lattices, that is, quantum Balgebras are not common basic of nonassociative fuzzy logics. So, we discuss more general implication algebras in this section. Deﬁnition 12. A basic implication algebra is a partially ordered set (X, ≤) with binary operation →, such that the following are satisﬁed for x, y, and z in X: (1) (2)
x ≤ y ⇒ z→x ≤ z→y; x ≤ y ⇒ y→z ≤ x→z.
(3) (4)
A basic implication algebra is considered to be normal, if it satisﬁes: for any x, y∈X, x→x = y→y; for any x, y∈X, x ≤ y ⇐⇒ x→y = e, where e = x→x = y→y.
We can verify that the following results are true (the proofs are omitted). Proposition 11. Let (X, →, ≤) be a basic implication algebra. Then, for all x, y, z∈X, (1) (2) (3) (4)
x ≤ y ⇒ y→x ≤ x→x ≤ x→y; x ≤ y ⇒ y→x ≤ y→y ≤ x→y; x ≤ y and u ≤ v ⇒ y→u ≤ x→v; x ≤ y and u ≤ v ⇒ v→x ≤ u→y.
Proposition 12. Let (X, →, ≤, e) be a normal basic implication algebra. Then for all x, y, z∈X, (1) (2) (3) (4)
x→x = e; x→y = y→x = e ⇒ x = y; x ≤ y ⇒ y→x ≤ e; if e is unit (that is, for all x in X, e→x = x), then e is a maximal element (that is, e ≤ x ⇒ e = x).
Proposition 13. (1) If (X, →, , ≤) is a a quantum Balgebra, then (X, →, ≤) and (X, , ≤) are basic implication algebras; (2) If (A, →, , ≤, 1) is a pseudoBCI algebra, then (A, →, ≤, 1) and (A, , ≤, 1) are = normal basic implication algebras with unit 1; (3) If (A, ∨, ∧, , →, 0, 1) is a nonassociative residuated lattice, then (A, →, ≤, 1) is a normal basic implication algebra. The following example shows that element e may not be a unit. Example 5. Let X = {a, b, c, d, 1}. Deﬁne a ≤ b ≤ c ≤ d ≤ 1 and operation → on X as per the following Cayley Table 7. Then, X is a normal basic implication algebra in which element 1 is not a unit. (X, →, ≤) is not a commutative quantum Balgebra, since c = 1 → c b = ( c → d ) → (1 → d ). Table 7. Cayley table of operation →.
→
a
b
c
d
1
a b c d 1
1 d d b b
1 1 d c b
1 1 1 d c
1 1 1 1 b
1 1 1 1 1
The following example shows that element e may be not maximal. 252
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Example 6. Let X = {a, b, c, d, 1}. Deﬁne a ≤ b ≤ c ≤ d, a ≤ b ≤ c ≤ 1 and operation → on X as per the following Cayley Table 8. Then, X is a normal basic implication algebra, and element 1 is not maximal and is not a unit. Table 8. Cayley table of operation →.
→
a
b
c
d
1
a b c d 1
1 c c a a
1 1 c c b
1 1 1 a b
1 1 1 1 c
1 1 1 c 1
Deﬁnition 13. A nonempty subset F of basic implication algebra (X, →, ≤) is called a ﬁlter of X if it satisﬁes: (1) (2) (3) (4) (5)
F is an upper set of X, that is, x∈F and x ≤ y∈X =⇒ y∈F; for all x∈F, x→x∈F; x∈F, y∈X, x→y∈F =⇒ y∈F; x∈X, y→z∈F =⇒ (x→y)→(x→z)∈F; x∈X, y→z∈F =⇒ (z→x)→(y→x)∈F.
(6)
For normal basic implication algebra (X, →, ≤, e), a ﬁlter F of X is considered to be regular, if it satisﬁes: x∈X, (x→y)→e∈F and (y→z)→e∈F =⇒ (x→z)→e∈F.
Proposition 14. Let (X, →, ≤, e) be a normal basic implication algebra and F ⊆ X. Then, F is a ﬁlter of X if and only if it satisﬁes: (1) (2) (3) (4)
e∈F; x∈F, y∈X, x→y∈F =⇒ y∈F; x∈X, y→z∈F =⇒ (x→y)→(x→z)∈F; x∈X, y→z∈F =⇒ (z→x)→(y→x)∈F.
Obviously, if e is the maximal element of normal basic implication algebra (X, →, ≤, e), then any ﬁlter of X is regular. Theorem 3. Let X be a basic implication algebra and F a ﬁlter of X. Deﬁne binary ≈F on X as follows: x≈F y ⇐⇒ x→y∈F and y→x∈F, where x, y∈X. Then (1) (2)
≈F is a equivalent relation on X; ≈F is a congruence relation on X, that is, x≈F y =⇒ (z→x) ≈F (z→y), (x→z) ≈F (y→z), for all z∈X.
Proof (1) ∀x∈X, from Deﬁnition 13 (2), x→x∈F, thus x ≈F x. Moreover, ∀x, y∈X, if x ≈F y, then y ≈F x. If x ≈F y and y ≈F z. Then x→y∈F, y→x∈F, y→z∈F, and z→y∈F. Applying Deﬁnition 13 (4) and (5), we have (x→y)→(x→z)∈F, (z→y)→(z→x)∈F. From this and Deﬁnition 13 (3), we have x→z∈F, z→x∈F. Thus, x ≈F z. Hence, ≈F is a equivalent relation on X.
253
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(2) Assume x ≈F y. By the deﬁnition of bianary relation ≈F , we have x→y∈F, y→x∈F. Using Deﬁnition 13 (4), (z→x)→(z→y)∈F, (z→y)→(z→x)∈F. This means that (z→x) ≈F (z→y). Moreover, using Deﬁnition 13 (5), we have (y→z)→(x→z)∈F, (x→z)→(y→z)∈F. Hence, (x→z) ≈F (y→z). Theorem 4. Let (X, →, ≤, e) be a normal basic implication algebra and F a regular ﬁlter of X. Deﬁne quotient operation → and binary relation ≤ on X/≈F as follows: [x]F →[y]F = [x]F →[y]F , ∀x, y∈X; [x]F ≤ [y]F ⇐⇒ [x]F →[y]F = [e]F , ∀x, y∈X. Then, (X/≈F , →, ≤, [e]F ) is a normal basic implication algebra, and (X, →, ≤, e) ∼ (X/≈F , →, ≤, [e]F ). Proof. Firstly, we prove that binary relation ≤ on X/≈F is a partial order. (1) ∀x∈X, obviously, [x]F ≤ [x]F . (2) Assume that [x]F ≤ [y]F and [y]F ≤ [x]F , then [x]F →[y]F = [x→y]F = [e]F , [y]F →[x]F = [y→x]F = [e]F . It follows that e→(x→y)∈F, e→(y→x)∈F. Applying Proposition 14 (1) and (2), we get that (x→y)∈F and (y→x)∈F. This means that [x]F = [y]F . (3) Assume that [x]F ≤ [y]F and [y]F ≤ [z]F , then [x]F →[y]F = [x→y]F = [e]F , [y]F →[z]F = [y→z]F = [e]F . Using the deﬁnition of equivalent relation ≈F , we have e→(x→y)∈F, (x→y)→e∈F; e→(y→z)∈F, (y→z)→e∈F. From e→(x→y)∈F and e→(y→z)∈F, applying Proposition 14 (1) and (2), (x→y)∈F and (y→z)∈F. By Proposition 14 (4), (x→y)→(x→z)∈F. It follows that (x→z)∈F. Hence, (x→x)→(x→z)∈F, by Proposition 14 (4). Therefore, e→(x→z) = (x→x)→(x→z)∈F. Moreover, from (x→y)→e∈F and (y→z)→e∈F, applying regularity of F and Deﬁnition 13 (6), we get that (x→z)→e∈F. Combining the above e→(x→z)∈F and (x→z)→e∈F, we have x→z ≈F e, that is, [x→z]F = [e]F . This means that [x]F ≤ [z]F . It follows that the binary relation ≤ on X/≈F is a partially order. Therefore, applying Theorem 3, we know that (X/≈F →, ≤, [e]F ) is a normal basic implication algebra, and (X, →, ≤, e) ∼ (X/≈F →, ≤, [e]F ) in the homomorphism mapping f : X→X/≈F ; f (x) = [x]F . Example 7. Let X = {a, b, c, d, 1}. Deﬁne operations → on X as per the following Cayley Table 9, and the order binary on X is deﬁned as follows: a ≤ b ≤ c ≤ 1, b ≤ d ≤ 1. Then (X, →, ≤, 1) is a normal basic implication algebra (it is not a quantum Balgebra). Denote F = {1}, then F is regular ﬁlters of X, and the quotient algebras (X, →, ≤, 1) is isomorphism to (X/≈F , →, [1]F ).
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Table 9. Cayley table of operation →.
→
a
b
c
d
1
a b c d 1
1 d b a a
1 1 d c b
1 1 1 c c
1 1 d 1 d
1 1 1 1 1
Example 8. Denote X = {a, b, c, d, 1}. Deﬁne operations → on X as per the following Cayley Table 10, and the order binary on X is deﬁned as follows: a ≤ b ≤ c ≤ 1, b ≤ d ≤ 1. Then (X, →, ≤, 1) is a normal basic implication algebra (it is not a quantum Balgebra). Let F = {1, d}, then F is a regular ﬁlters of X, and the quotient algebras (X/≈F , →, [1]F ) is presented as the following Table 11, where X/≈F = {{a}, {b, c}, [1]F = {1, d}}. Moreover, (X, →, ≤, 1) ∼ (X/≈F →, [1]F ). Table 10. Cayley table of operation →.
→
a
b
c
d
1
a b c d 1
1 c b a a
1 1 d c b
1 1 1 c c
1 1 d 1 d
1 1 1 1 1
Table 11. Quotient algebra (X/≈F , →, [1]F ).
→
{a}
{b,c}
[1]F
{a} {b,c} [1]F
[1]F {b,c} {a}
[1]F [1]F {b,c}
[1]F [1]F [1]F
5. Conclusions In this paper, we introduced the notion of a qﬁlter in quantum Balgebras and investigated quotient structures; by using qﬁlters as a corollary, we obtained quotient pseudoBCI algebras by their ﬁlters. Moreover, we pointed out that the concept of quantum Balgebra does not apply to nonassociative fuzzy logics. From this fact, we proposed the new concept of basic implication algebra, and established the corresponding ﬁlter theory and quotient algebra. In the future, we will study in depth the structural characteristics of basic implication algebras and the relationship between other algebraic structures and uncertainty theories (see References [33–36]). Moreover, we will consider the applications of qﬁlters for Gentzel’s sequel calculus. Author Contributions: X.Z. initiated the research and wrote the draft. B.R.A., Y.B.J., and X.Z. completed the ﬁnal version. Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 61573240). Conﬂicts of Interest: The authors declare no conﬂict of interest.
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SS symmetry Article
Fuzzy Normed Rings 2, * Aykut Emniyet 1 and Memet Sahin ¸ 1 2
*
Department of Mathematics, Osmaniye Korkut Ata University, 8000 Osmaniye, Turkey; [email protected] Department of Mathematics, Gaziantep University, 27310 Gaziantep, Turkey Correspondence: [email protected]; Tel.: +905432182646
Received: 12 September 2018; Accepted: 8 October 2018; Published: 16 October 2018
Abstract: In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our deﬁnition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We deﬁne fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples. Keywords: Fuzzy sets; ring; normed space; fuzzy normed ring; fuzzy normed ideal
1. Introduction Normed rings attracted attention of researchers after the studies by Naimark [1], a generalization of normed rings [2] and commutative normed rings [3]. Naimark deﬁned normed rings in an algebraic fashion, while Gel’fand addressed them as complex Banach spaces and introduced the notion of commutative normed rings. In Reference [4], Jarden deﬁned the ultrametric absolute value and studied the properties of normed rings in a more topological perspective. During his invaluable studies, Zadeh [5] presented fuzzy logic theory, changing the scientiﬁc history forever by making a modern deﬁnition of vagueness and using the sets without strict boundaries. As, in almost every aspect of computational science, fuzzy logic also became a convenient tool in classical algebra. Zimmermann [6] made signiﬁcant contributions to the fuzzy set theory. Mordeson, Bhutani, and Rosenfeld [7] deﬁned fuzzy subgroups, Liu [8], Mukherjee, and Bhattacharya [9] examined normal fuzzy subgroups. Liu [8] also discussed fuzzy subrings and fuzzy ideals. Wang, Ruan and Kerre [10] studied fuzzy subrings and fuzzy rings. Swamy and Swamy [11] deﬁned and proved major theorems on fuzzy prime ideals of rings. Gupta and Qi [12] are concerned with Tnorms, Tconorms and Toperators. In this study, we use the deﬁnitions of Kolmogorov, Silverman, and Formin [13] on linear spaces and norms. Uluçay, S¸ ahin, and Olgun [14] worked out on normed ZModules and also on soft normed rings [15]. S¸ ahin, Olgun, and Uluçay [16] deﬁned normed quotient rings while S¸ ahin and Kargın [17] presented neutrosophic triplet normed space. In Reference [18], Olgun and S¸ ahin investigated ﬁtting ideals of the universal module and while Olgun [19] found a method to solve a problem on universal modules. S¸ ahin and Kargin proposed neutrosophic triplet inner product [20] and Florentin, S¸ ahin, and Kargin introduced neutrosophic triplet Gmodule [21]. S¸ ahin and et al deﬁned isomorphism theorems for soft Gmodule in [22]. Fundamental homomorphism theorems for neutrosophic extended triplet groups [23] were introduced by Mehmet, Moges, and Olgun in 2018. In Reference [24], Bal, Moges, and Olgun introduced neutrosophic triplet cosets and quotient groups, and deal with its application areas in neutrosophic logic. This paper anticipates a normed ring on R and fuzzy rings are deﬁned in the previous studies. Now, we use that norm on fuzzy sets, hence a fuzzy norm is obtained and by deﬁning our fuzzy norm on fuzzy rings, we get fuzzy normed rings in this study. The organization of this paper is as follows. Symmetry 2018, 10, 515; doi:10.3390/sym10100515
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In Section 2, we give preliminaries and fuzzy normed rings. In Section 3, consists of further deﬁnitions and relevant theorems on fuzzy normed ideals of a normed ring. Fuzzy normed prime and fuzzy normed maximal ideals of a normed ring are introduced in Section 4. The conclusions are summarized in Section 5. 2. Preliminaries In this section, deﬁnition of normed linear space, normed ring, Archimedean strict Tnorm and concepts of fuzzy sets are outlined. Deﬁnition 1. [13] A functional $$ deﬁned on a linear space L is said to be a norm (in L) if it has the following properties: N1: $ x $ ≥ 0 for all x ∈ L, where $ x $ = 0 if and only if x = 0; N2: $α · x $ = α.$ x $; (and hence $ x $ = $− x $), for all x ∈ L and for all α; N3: Triangle inequality: $ x + y$ ≤ $ x $ + $y$ for all x, y ∈ L. A linear space L, equipped with a norm is called a normed linear space. Deﬁnition 2. [3] A ring A is said to be a normed ring if A possesses a norm $$, that is, a nonnegative realvalued function $$ : A → R such that for any a, b ∈ A, 1. 2. 3. 4.
$ a$ = 0 ⇔ a = 0 , $ a + b $ ≤ $ a $ + $ b $, $− a$ = $ a$, (and hence $1 A $ = 1 = $−1$ if identity exists), and $ ab$ ≤ $ a$$b$.
Deﬁnition 3. [12] Let ∗ : [0, 1] × [0, 1] → [0, 1] . ∗ is an Archimedean strict Tnorm iff for all x, y, z ∈ [0, 1]: (1) (2) (3) (4) (5) (6)
∗ is commutative and associative, that is, ∗ ( x, y) = ∗ (y, x ) and ∗ ( x, ∗ (y, z)) = ∗ (∗ ( x, y), z), ∗ is continuous, ∗ ( x, 1) = x , ∗ is monotone, which means ∗ ( x, y) ≤ ∗ ( x, z) if y ≤ z, ∗ ( x, x ) < x for x ∈ (0, 1), and when x < z and y < t, ∗ ( x, y) < ∗ (z, t) for all x, y, z, t ∈ (0, 1).
For convenience, we use the word tnorm shortly and show it as x ∗ y instead of ∗ ( x, y) . Some examples of tnorms are x ∗ y = min{ x, y}, x ∗ y = max{ x + y − 1, 0} and x ∗ y = x.y. Deﬁnition 4. [12] Let % : [0, 1] × [0, 1] → [0, 1] . ∗ is an Archimedean strict Tconorm iff for all x, y, z ∈ [0, 1]: (1) (2) (3) (4) (5) (6)
% is commutative and associative, that is, % ( x, y) = % (y, x ) and % ( x, % (y, z)) = % (% ( x, y), z), % is continuous, % ( x, 0) = x , % is monotone, which means % ( x, y) ≤ % ( x, z) if y ≤ z, % ( x, x ) > x for x ∈ (0, 1), and when x < z and y < t, % (z, t) < % ( x, y) for all x, y, z, t ∈ (0, 1).
For convenience, we use the word snorm shortly and show it as x % y instead of % ( x, y) . Some examples of snorms are x % y = max{ x, y}, x ∗ y = min{ x + y, 1} and x % y = x + y − x.y. Deﬁnition 5. [6] The fuzzy set B on a universal set X is a set of ordered pairs B = {( x, μ B ( x ) : x ∈ X )}
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Here, μ B ( x ) is the membership function or membership grade of x in B. For all x ∈ X, we have 0 ≤ μ B ( x ) ≤ 1. If x ∈ / B, μ B ( x ) = 0, and if x is entirely contained in B, μ B ( x ) = 1. The membership grade of x in B is shown as B( x ) in the rest of this paper. Deﬁnition 6. [6] For the fuzzy sets A and B, the membership functions of the intersection, union and complement are deﬁned pointwise as follows respectively:
( A ∩ B)( x ) = min{ A( x ), B( x )}, ( A ∪ B)( x ) = max{ A( x ), B( x )}, A ( x ) = 1 − A ( x ). Deﬁnition 7. [10] Let ( R, +, .) be a ring and F ( R) be the set of all fuzzy subsets of R. As A ∈ F ( R), ∧ is the fuzzy intersection and ∨ is the fuzzy union functions, for all x, y ∈ R, if A satisﬁes (1) A( x − y) ≥ A( x ) ∧ A(y) and (2) A( x.y) ≥ A( x ) ∧ A(y) then A is called a fuzzy subring of R. If A is a subring of R for all a ∈ A, then A is itself a fuzzy ring. Deﬁnition 8. [11] A nonempty fuzzy subset A of R is said to be an ideal (in fact a fuzzy ideal) if and only if, for any x, y ∈ R, A( x − y) ≥ A( x ) ∧ A(y) and A( x.y) ≥ A( x ) ∨ A(y). Note: The fuzzy operations of the fuzzy subsets A, B ∈ F ( R) on the ring R can be extended to the operations below by tnorms and snorms: For all z ∈ R, ( A + B)(z) = % ( A( x ) ∗ B(y)); x +y=z
( A − B)(z) =
% ( A( x ) ∗ B(y));
x −y=z
( A.B)(z) = % ( A( x ) ∗ B(y)). x.y=z
3. Fuzzy Normed Rings and Fuzzy Normed Ideals In this section, there has been deﬁned the fuzzy normed ring and some basic properties related to it. Throughout the rest of this paper, R is the set of real numbers, R will denote an associative ring with identity, NR is a normed ring and F ( X ) is the set of all fuzzy subsets of the set X. Deﬁnition 9. Let ∗ be a continuous tnorm and % a continuous snorm, NR a normed ring and let A be a fuzzy set. If the fuzzy set A = {( x, μ A ( x )) : x ∈ NR} over a fuzzy normed ring F ( NR) satisfy the following conditions then A is called a fuzzy normed subring of the normed ring ( NR, +, .): For all x, y ∈ NR, (i) (ii)
A( x − y) ≥ A( x ) ∗ A(y) A( x.y) ≥ A( x ) ∗ A(y).
Let 0 be the zero of the normed ring NR. For any fuzzy normed subring A and for all x ∈ NR, we have A( x ) ≤ A(0), since A( x − x ) ≥ A( x ) ∗ A( x ) ⇒ A(0) ≥ A( x ) . Example 1. Let A fuzzy set and R = ( Z, +, .) be the ring of all integers. f : A → F ( NR( Z )) where, for any a ∈ A and x ∈ Z, ( A f (x) =
0 if x is odd 1 a if x is even
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Corresponding tnorm (∗) and tconorm (%) are deﬁned as a ∗ b = min{ a, b}, a % b = max { a, b}; then, A is a fuzzy set as well as a fuzzy normed ring over [( Z, +, .), A]. Lemma 1. A ∈ F ( NR) is a fuzzy normed subring of the normed ring NR if and only if A − A ⊆ A and A.A ⊆ A. Proof. Let A be a fuzzy normed subring of NR. By [10], it is clear that A is a fuzzy group under addition and so A − A ⊆ A. Also for all z ∈ NR,
( A.A)(z) = % ( A( x ) ∗ A(y)) ≤ % A( xy) = A(z) ⇒ A.A ⊆ A x.y=z
x.y=z
Now we suppose A − A ⊆ A and A.A ⊆ A. For all x, y ∈ NR, A( x − y) ≥ ( A − A)( x − y) =
%
s−t= x −y
( A(s) ∗ A(t)) ≥ A( x ) ∗ A(y).
Similarly, A( xy) ≥ ( A.A)( xy) =
% ( A(s) ∗ A(t)) ≥ A( x ) ∗ A(y).
st= xy
Thus, A is a fuzzy normed subring of NR. Lemma 2. i. ii.
Let A be a fuzzy normed subring of the normed ring NR and let f : NR → NR be a ring homomorphism. Then, f ( A) is a fuzzy normed subring of NR. Let f : NR → NR be a normed ring homomorphism. If B is a fuzzy normed subring of NR, then f −1 ( B) is a fuzzy normed subring of NR.
Proof. (i) Take u, v ∈ NR. As f is onto, there exists x, y ∈ NR such that f ( x ) = u and f (y) = v. So,
( f ( A))(u) ∗ ( f ( A))(v) = = ≤ ≤ = =
%
f ( x )=u
A( x ) ∗
f (y)=v
A(y)
%
( A( x ) ∗ A(y))
%
( A( x − y)) (as A is a fuzzy normed subring of NR)
f ( x )=u, f (y)=v f ( x )=u, f (y)=v
%
f ( x )− f (y)=u−v
%
f ( x −y)=u−v
%
%
f (z)=u−v
( A( x − y))
( A( x − y)) (since f is a homomorphism)
A(z)
= ( f ( A))(u − v). Similarly, it is easy to see that
( f ( A))(u.v) ≥ ( f ( A)(u) ∗ f ( A)(v)). Therefore, f ( A) is a fuzzy normed subring of NR. (ii) Proof is straightforward and similar to the proof of (i). Deﬁnition 10. Let A1 and A2 be two fuzzy normed rings over the normed ring NR. Then A1 is a fuzzy normed subring of A2 if A1 ( x ) ≤ A2 ( x ) for all x ∈ NR. 261
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Deﬁnition 11. Let NR be a normed ring, A ∈ F ( NR) and let A = ∅. If for all x, y ∈ NR (i) (ii)
A( x − y) ≥ A( x ) ∗ A(y) and A( x.y) ≥ A(y) ( A( x.y) ≥ A( x )),
then A is called a fuzzy left (right) normed ideal of NR. Deﬁnition 12. If the fuzzy set A is both a fuzzy normed right and a fuzzy normed left ideal of NR, then A is called a fuzzy normed ideal of NR; i.e., if for all x, y ∈ NR (i) (ii)
A( x − y) ≥ A( x ) ∗ A(y) and A( x.y) ≥ A( x ) % A(y),
then A ∈ F ( NR) is a fuzzy normed ideal of NR. Remark 1. Let the multiplicative identity of NR (if exists) be 1 NR . As A( x.y) ≥ A( x ) % A(y) for all x, y ∈ NR, A( x.1 NR ) ≥ A( x ) % A(1 NR ) and therefore for all x ∈ NR, A( x ) ≥ A(1 NR ). Example 2. Let A and B be two (fuzzy normed left, fuzzy normed right) ideals of a normed ring NR. Then, A ∩ B is also a (fuzzy normed left, fuzzy normed right) ideal of NR. Solution: Let x, y ∈ NR.
( A ∩ B)( x − y) = min{ A( x − y), B( x − y)} ≥ min{ A( x ) ∗ A(y), B( x ) ∗ B(y)} ≥ min{( A ∩ B)( x ), ( A ∩ B)(y)}. On the other hand, as A and B are fuzzy normed left ideals, using A( x.y) ≥ A(y) and B( x.y) ≥ B(y) we have
( A ∩ B)( x.y) = min{ A( x.y), B( x.y)} ≥ min{ A(y), B(y)} = ( A ∩ B)(y). So A ∩ B is a fuzzy normed left ideal. Similarly, it is easy to show that A ∩ B is a fuzzy normed right ideal. As a result A ∩ B is an fuzzy normed ideal of NR. Example 3. Let A be a fuzzy ideal of NR. The subring A0 = { x : μ A ( x ) = μ A (0 NR )} is a fuzzy normed ideal of NR, since for all x ∈ NR, A0 ( x ) ≤ A0 (0). Theorem 1. Let A be a fuzzy normed ideal of NR, X = { a1 , a2 , . . . , am } ⊆ NR, x, y ∈ NR and let FN ( X ) be the fuzzy normed ideal generated by the set X in NR. Then, (i)
w ∈ FN ( X ) ⇒ A(w) ≥
∗ ( A( ai )) ,
1≤ i ≤ m
(ii) x ∈ (y) ⇒ A( x ) ≥ A(y) , (iii) A(0) ≥ A( x ) and (iv) if 1 is the multiplicative identity of NR, then A( x ) ≥ A(1). Proof. (ii), (iii), and (iv) can be proved using (i). The set FN ( X ) consists of the ﬁnite sums in the form ra + as + uav + na where a ∈ X, r, s, u, v ∈ NR and n is an integer. Let w ∈ FN ( X ). So there exists an integer n and r, s, u, v ∈ NR such that w = rai + ai s + uai v + nai where 1 ≤ i ≤ m. As A is a fuzzy normed ideal, A(rai + ai s + uai v + nai ) ≥ A(rai ) ∗ A( ai s) ∗ A(uai v) ∗ A(nai ) ≥ A( ai ).
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Therefore A(w) ≥
∗ ( A( ai )).
1≤ i ≤ m
4. Fuzzy Normed Prime Ideal and Fuzzy Normed Maximal Ideal In this section, fuzzy normed prime ideal and fuzzy normed maximal ideal are outlined. Deﬁnition 13. Let A and B be two fuzzy subsets of the normed ring NR. We deﬁne the operation A ◦ B as follows: ( % ( A(y) ∗ B(z)) , if x can be deﬁned as x = yz x =yz A ◦ B( x ) = 0 , otherwise . If the normed ring NR has a multiplicative inverse, namely if NR.NR = NR, then the second case does not occur. Lemma 3. If A and B are a fuzzy normed right and a fuzzy normed left ideal of a normed ring NR, respectively, A ◦ B ⊆ A ∩ B and hence ( A ◦ B)( x ) ≤ ( A ∩ B)( x ) for all x ∈ NR. Proof. It is shown in Example 2 that if A and B are fuzzy normed left ideals of NR, then A ∩ B is also a fuzzy normed left ideal. Now, let A and B be a fuzzy normed right and a fuzzy normed left ideal of NR, respectively. If A ◦ B( x ) = 0, the proof is trivial. Let ( A ◦ B)( x ) = % ( A(y) ∗ B(z)). x =yz
As A is a fuzzy normed right ideal and B is a fuzzy normed left ideal, we have A(y) ≤ A(yz) = A( x ) and B(y) ≤ B(yz) = B( x ) Thus,
( A ◦ B)( x ) = % ( A(y) ∗ B(z)) x =yz
≤ min( A( x ), B( x )) = ( A ∩ B)( x ) Deﬁnition 14. Let A and B be fuzzy normed ideals of a normed ring NR and let FNP be a nonconstant function, which is not an ideal of NR. If A ◦ B ⊆ FNP ⇒ A ⊆ FNP or B ⊆ FNP, then FNP is called a fuzzy normed prime ideal of NR. Example 4. Show that if the fuzzy normed ideal I ( I = NR) is a fuzzy normed prime ideal of NR, then the characteristic function λ I is also a fuzzy normed prime ideal. Solution: As I = NR, λ I is a nonconstant function on NR. Let A and B be two fuzzy normed ideals on NR such that A ◦ B ⊆ λ I , but Aλ I and Bλ I . There exists x, y ∈ NR such that A( x ) ≤ λ I ( x ) and B(y) ≤ λ I (y). In this case, A( x ) = 0 and B(y) = 0, but λ I ( x ) = 0 and λ I (y) = 0. Therefore
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x∈ / I, y ∈ / I. As I is a fuzzy normed prime ideal, there exists an r ∈ NR, such that xry ∈ / I. This is obvious, because if I is fuzzy normed prime, A ◦ B( xry) ⊆ I ⇒ A( x ) ⊆ I or B(ry) ⊆ I and therefore as ( NRxNR)( NRryNR) = ( NRxNR)( NRyNR) ⊆ I, we have either NRxNR ⊆ I or NRyNR ⊆ I. Assume NRxNR ⊆ I. Then xxx = ( x )3 ∈ I ⇒ x ⊆ I , but this contradicts with the fact that λ I ( x ) = 0. Now let a = xry. λ I ( a) = 0. Thus, A ◦ B( a) = 0. On the other hand, A ◦ B( a)
= % ( A(c) ∗ B(d)) a=cd
≥ A( x ) ∗ B(ry) ≥ A( x ) ∗ B(y) ≥ 0 ( as A( x ) = 0 and B(y) = 0). This is a contradiction, since A ◦ B( a) = 0. Therefore if A and B are fuzzy normed ideals of a normed ring NR, then A ◦ B ⊆ λ I ⇒ A ⊆ λ I or B ⊆ λ I . As a result, the characteristic function λ I is a fuzzy normed prime ideal. Theorem 2. Let FNP be a fuzzy normed prime ideal of a normed ring NR. The ideal deﬁned by FNP0 = { x : x ∈ NR, FNP( x ) = FNP(0)} and is also a fuzzy normed prime ideal of NR. Proof: Let x, y ∈ FNP0 . As FNP is an fuzzy normed ideal, FNP( x − y) ≥ FNP( x ) ∗ FNP(y) = FNP(0). On the other hand, by Theorem 1, we have FNP(0) ≥ FNP( x − y). So, FNP( x − y) = FNP(0) and x − y ∈ FNP0 . Now, let x ∈ FNP0 and r ∈ NR. In this case, FNP(rx ) ≥ FNP( x ) = FNP(0) and thus FNP(rx ) = FNP(0). Similarly, FNP( xr ) = FNP(0). Now, for all x ∈ FNP0 and r ∈ NR, rx, xr ∈ FNP0 . Therefore, FNP0 is a fuzzy normed ideal of NR. Let I and J be two ideals of NR, such that I J ⊆ FNP0 . Now, we deﬁne fuzzy normed ideals A = FNP0 λ I and B = FNP0 λ J . We will show that ( A ◦ B)( x ) ≤ FNP( x ) for all x ∈ NR. Assume ( A ◦ B)( x ) = 0. Recall A ◦ B = % ( A(y) ∗ B(z)), so we only need to take the cases of A(y) ∗ B(z) = 0 under consideration. However, x =yz
in all these cases, A(y) = FNP(0) or A(y) = 0 and similarly B(z) = FNP(0) or B(z) = 0 and hence A(y) = B(z) = FNP(0). Now, λ I (y) = 1 and λ J (z) = 1 implies y ∈ I, z ∈ J and x ∈ I J ⊆ FNP0 . Thus, FNP( x ) = FNP(0) and for all x ∈ NR, we get ( A ◦ B)( x ) ≤ FNP( x ). As FNP is a fuzzy normed prime ideal and A and B are fuzzy normed ideals, either A ⊆ FNP or B ⊆ FNP. Assume A = FNP0 λ I ⊆ FNP. We need to show that I ⊆ FNP0 . Let IFNP. Then, there exists an a ∈ I, such that a ∈ / FNP0 ; i.e., FNP( a) = FNP(0). It is evident that FNP(0) ≥ FNP( a). Thus, FNP( a)FNP( a) and this is a contradiction to the assumption A ⊆ FNP. So, I ⊆ FNP0 . Similarly, one can show that B ⊆ FNP and J ⊆ FNP0 . Thus, FNP0 is a fuzzy normed prime ideal. Deﬁnition 15. Let A be a fuzzy normed ideal of a normed ring NR. If A is nonconstant and for all fuzzy normed ideals B of NR, A ⊆ B implies A0 = B0 or B = λ NR , A is called a fuzzy normed maximal ideal of the normed ring NR. Fuzzy normed maximal left(right) ideals are deﬁned similarly. Example 5. Let A be a fuzzy normed maximal left (right) ideal of a normed ring NR. Then, A0 = { x ∈ NR : A( x ) = A(0)} is a fuzzy normed maximal left (right) ideal of NR. Theorem 3. If A is a fuzzy normed left(right) maximal ideal of a normed ring NR, then A(0) = 1. Proof. Assume A(0) = 1. Let A(0) < t < 1 and let B be a fuzzy subset of NR such that B( x ) = t for all x ∈ NR. B is trivially an ideal of NR. Also it is easy to verify that A ⊂ B, B = λ NR and B0 = { x ∈ NR : B( x ) = B(0)} = NR. But, despite the fact that A ⊂ B, A0 = B0 and B = λ NR is a contradiction to the fuzzy normed maximality of A. Thus, A(0) = 1.
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5. Conclusions In this paper, we deﬁned a fuzzy normed ring. Here we examine the algebraic properties of fuzzy sets in ring structures. Some related notions, e.g., the fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal and fuzzy normed maximal ideal are proposed. We hope that this new concept will bring a new opportunity in research and development of fuzzy set theory. To extend our work, further research can be done to study the properties of fuzzy normed rings in other algebraic structures such as fuzzy rings and fuzzy ﬁelds. Author Contributions: All authors contributed equally. Acknowledgments: We thank Vakkas Uluçay for the arrangement. Conﬂicts of Interest: The authors declare no conﬂict of interest.
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SS symmetry Article
Invariant Graph Partition Comparison Measures Fabian Ball * and Andreas GeyerSchulz Karlsruhe Institute of Technology, Institute of Information Systems and Marketing, Kaiserstr. 12, 76131 Karlsruhe, Germany; [email protected] * Correspondence: [email protected]; Tel.: +4972160848404 Received: 10 September 2018; Accepted: 11 October 2018; Published: 15 October 2018
Abstract: Symmetric graphs have nontrivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived. Keywords: graph partitioning; graph clustering; invariant measures; partition comparison; ﬁnite automorphism groups; graph automorphisms
1. Introduction Partition comparison measures are routinely used in a variety of tasks in cluster analysis: ﬁnding the proper number of clusters, assessing the stability and robustness of solutions of cluster algorithms, comparing different solutions of randomized cluster algorithms or comparing optimal solutions of different cluster algorithms in benchmarks [1], or in competitions like the 10th DIMACS graphclustering challenge [2]. Their development has been for more than a century an active area of research in statistics, data analysis and machine learning. One of the oldest and still very wellknown measure is the one of Jaccard [3]; more recent approaches were by Horta and Campello [4] and Romano et al. [5]. For an overview of many of these measures, see Appendix B. Besides the need to compare clustering partitions, there is an ongoing discussion of what actually are the best clusters [6,7]. Another problem often addressed is how to measure cluster validity [8,9]. However, the comparison of graph partitions leads to new challenges because of the need to handle graph automorphisms properly. The following small example shows that standard partition comparison measures have unexpected results when applied to graph partitions: in Figure 1, we show two different ways of partitioning the cycle graph C4 (Figure 1a,d). Partitioning means grouping the nodes into nonoverlapping clusters. The nodes are arbitrarily labeled with 1 to 4 (Figure 1b,e), and then, there are four possibilities of relabeling the nodes so that the edges stay the same. One possibility is relabeling 1 by 2, 2 by 3, 3 by 4 and 4 by 1, and the images resulting from this relabeling are shown in Figure 1c,f. The relabeling corresponds to a counterclockwise rotation of the graph by 90◦ , and formal details are given in Section 2. The effects of this relabeling on the partitions P1 and Q1 are different: 1. 2.
Partition P1 = {{1, 2}, {3, 4}} is mapped to the structurally equivalent partition P2 = {{1, 4}, {2, 3}}. Partition Q1 = {{1, 3}, {2, 4}} is mapped to the identical partition Q2 .
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1
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(b) P1
(a)
(c) P2
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Figure 1. Two structurally different partitions of the cycle graph C4 : grouping pairs of neighbors (a) and grouping pairs of diagonals (d). Equallycolored nodes represent graph clusters, and the choice of colors is arbitrary. Adding, again arbitrary, but ﬁxed, node labels impacts the node partitions and results in the failure to recognize the structural difference when comparing these partitions with partition comparison measures (see Table 1). The different images (b,c) (P1 = {{1, 2}, {3, 4}}, P2 = {{1, 4}, {2, 3}}) and (e,f) (Q1 = Q2 = {{1, 3}, {2, 4}}) emerge from the graph’s symmetry.
Table 1 illustrates the failure of partition comparison measures (here, the Rand Index (RI)) to recognize structural differences: 1. 2.
Because P1 and P2 are structurally equivalent, the RI should be one (as for Cases 1, 2 and 3) instead of 1/3. Comparisons of structurally different different partitions (Cases 4 and 5) and comparisons of structurally equivalent partitions (Case 6) should not result in the same value. 11 + N00 Table 1. The Rand index is RI = N11 + N N10 + N01 + N00 . N11 indicates the number of nodes that are in both partitions together in a cluster; N10 and N01 are the number of nodes that are together in a cluster in one partition, but not in the other; and N00 are the number of nodes that are in both partitions in different clusters. See Appendix B for the formal deﬁnitions. Partitions P1 and P2 are equivalent (yet not equal, denoted “∼”), and partitions Q1 and Q2 are identical (thus, also equivalent, denoted “=”). However, the comparison of the structurally different partitions (denoted “ =”) Pi and Q j yields the same result as the comparison between the equivalent partitions P1 and P2 . This makes the recognition of structural differences impossible.
Case
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One may argue that graphs in real applications contain symmetries only rarely. However, recent investigations of graph symmetries in real graph datasets show that a nonnegligible proportion of these graphs contain symmetries. MacArthur et al. [10] state that “a certain degree of symmetry is also ubiquitous in complex systems” [10] (p. 3525). Their study includes a small number of biological, technological and social networks. In addition, Darga et al. [11] studied automorphism groups in very large sparse graphs (circuits, road networks and the Internet router network), with up to ﬁve million nodes with eight million links with execution times below 10 s. Katebi et al. [12] reported symmetries in 268 of 432 benchmark graphs. A recent largescale study conducted by the authors of this article for approximately 1700 realworld graphs revealed that about three quarters of these graphs contain symmetries [13]. The rather frequent occurrence of symmetries in graphs and the obvious deﬁciencies of classic partition comparison measures demonstrated above have motivated our analysis of the effects of graph automorphisms on partition comparison measures. Our contribution has the following structure: Permutation groups and graph automorphisms are introduced in Section 2. The full automorphism group of the butterﬂy graph serves as a motivating example for the formal deﬁnition of stable partitions, stable with regard to the actions of the automorphism group of a graph. In Section 3, we ﬁrst provide a deﬁnition that captures the property that a measure is invariant with regard to the transformations in an automorphism group. Based on this deﬁnition, we ﬁrst give a simple proof by counterexample for each partition comparison measure in Appendix B, that these measures based on the comparison of two partitions are not invariant to the effects of automorphisms on partitions. The nonexistence of partition comparison measures for which the identity and the invariance axioms hold simultaneously is proven subsequently. In Section 4, we construct three families of invariant partition comparison measures by a twostep process: First, we deﬁne a pseudometric space by deﬁning equivalence classes of partitions as the orbit of a partition under the automorphism group Aut( G ). Second, the deﬁnitions of the invariant counterpart of a partition comparison measure are given: we deﬁne them as the computation of the maximum, the minimum and the average of the direct product of the two equivalence classes. The section also contains a proof of the equivalence of several variants of the computation of the invariant measures, which—by exploiting the group properties of Aut( G )—differ in the complexity of the computation. In Section 5, we introduce the decomposition of the measures into a structurally stable and unstable part, as well as upper bounds for instability. In Section 6, we present an application of the decomposition of measures for analyzing partitions of the Karate graph. The article ends with a short discussion, conclusion and outlook in Section 7. 2. Graphs, Permutation Groups and Graph Automorphisms We consider connected, undirected, unweighted and loopfree graphs. Let G = (V, E) denote a graph where V is a ﬁnite set of nodes and E is a set of edges. An edge is represented as {u, v} ∈ {{ x, y}  ( x, y) ∈ V × V ∧ x = y}. Nodes adjacent to u ∈ V (there exists an edge between u and those nodes) are called neighbors. A partition P of a graph G is a set of subsets Ci , i = 1, . . . , k of V with the * usual properties: (i) Ci ∩ Cj = ∅ (i = j), (ii) i Ci = V and (iii) Ci = ∅. Each subset is called a cluster, and it is identiﬁed by its labeled nodes. As a partition quality criterion, we use the wellknown modularity measure Q of Newman and Girvan [14] (see Appendix A). It is a popular optimization criterion for unsupervised graph clustering algorithms, which try to partition the nodes of the graph in a way that the connectivity within the clusters is maximized and the number of edges connecting the clusters is minimized. For a fast and efﬁcient randomized stateoftheart algorithm, see Ovelgönne and GeyerSchulz [15]. Partitions are compared by comparison measures, which are functions of the form m : P(V ) × P(V ) → R where P(V ) denotes the set of all possible partitions of the set V. A survey of many of these measures is given in Appendix B.
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A permutation on V is a bijection g : V → V. We denote permutations by the symbols f , g and h. Each permutation can be written in cycle form: for a permutation with a single cycle of length r, we write c = (v1 v2 . . . vr ). c maps vi to vi+1 (i = 1, . . . , r − 1), vr to v1 and leave all other nodes ﬁxed. Permutations with more than one cycle are written as a product of disjoint cycles (i.e., no two cycles have a common element). (vk ) means that the element vk remains ﬁxed, and for brevity, these elements are omitted. Permutations are applied from the right: The image of u under the permutation g is ug. The composition of g and h is h ◦ g, with ◦ being the permutation composition symbol. For brevity, h ◦ g is written as gh, so that u( gh) = (ug)h holds. Computer scientists call this a postﬁx notation; in preﬁx notation, we have h( g(u)). Often, we also ﬁnd u g , which we will use in the following. For k compositions g ◦ g ◦ g ◦ . . ., we write gk and g0 = id. A set of permutation functions forms a permutation group H, if the usual group axioms hold [16]: 1. 2. 3. 4.
Closure: ∀ g, h ∈ H : g ◦ h ∈ H Unit element: The identity function id ∈ H acts as the neutral element: ∀ g ∈ H : id ◦ g = g ◦ id = g Inverse element: For any g in H, the inverse permutation function g−1 ∈ H is the inverse of g: ∀ g ∈ H : g ◦ g−1 = g−1 ◦ g = id Associativity: The associative law holds: ∀ f , g, h ∈ H : f ◦ ( g ◦ h) = ( f ◦ g) ◦ h
If H1 is a subset of H and if H1 is a group, H1 is a subgroup of H (written H1 ≤ H). The set of all permutations of V is denoted by Sym(V ). Sym(V ) is a group, and it is called the symmetric group (see [17]). Sym(V ) ∼ Sym(V ) iff V  = V  with ∼ denoting isomorphism. A generator of a ﬁnite permutation group H is a subset of the permutations of H from which all permutations in H can be generated by application of the group axioms [18]. An action of H on V (H acts on V) is called the group action of a set [19] (p. 5): 1. 2.
uid = u, ∀u ∈ V (u g )h = u gh , ∀u ∈ V, ∀ g, h ∈ H
Groups acting on a set V also act on combinatorial structures deﬁned on V [20] (p. 149), for example the power set 2V , the set of all partitions P(V ) or the set of graphs G (V ). We denote combinatorial structures as capital calligraphic letters; in the following, only partitions (P ) are of interest because they are the results of graph cluster algorithms. The action of a permutation g on a combinatorial structure is performed by pointwise application of g. For instance, for P , the image of g is P g = {{u g  u ∈ C }  C ∈ P }. Let H be a permutation group. When H acts on V, a node u is mapped by the elements of H onto other nodes. The set of these images is called the orbit of u under H: $ # u H = uh  h ∈ H . The group of permutations Hu that ﬁxes u is called the stabilizer of u under H: Hu = {h ∈ H  uh = u}. The orbit stabilizer theorem is given without proof [16]. It links the order of a permutation group with the cardinality of an orbit and the order of the stabilizer: Theorem 1. The relation:
 H  = u H  ·  Hu  holds. The action of H on V induces an equivalence relation on the set: for u1 , u2 ∈ V, let u1 ∼ u2 iff there exists h ∈ H so that u1 = u2h . All elements of an orbit are equivalent, and the orbits of a group 269
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partition the set V. An orbit of length one (in terms of set cardinality) is called trivial. Analogously, for a partition P , the deﬁnition is: Deﬁnition 1. The image of the action of H on a partition P (or the orbit of P under H) is the set of all equivalent partitions of partition P under H # $ PH = Ph  h ∈ H . A graph automorphism f is a permutation that preserves edges, i.e., {u f , v f } ∈ E ⇔ {u, v} ∈ E, ∀u, v ∈ V. The automorphism group of a graph contains all permutations of vertices that map edges to edges and nonedges to nonedges. The automorphism group of G is deﬁned as: $ # Aut( G ) = f ∈ Sym(V )  E f = E $ # where E f = {u f , v f }  {u, v} ∈ E . Of course, Aut( G ) ≤ Sym(V ). Example 1. Let Gb f be the butterﬂy graph (Figure 2, e.g., Erd˝os et al. [21], Burr et al. [22]) whose full automorphism group is given in Table 2 (ﬁrst column). The permutation (2 5) is not an automorphism, because it does not preserve the edges from 1 to 2 and from 5 to 4. The butterﬂy graph has the two orbits {1, 2, 4, 5} and {3}. The group H = {id, g1 , g2 , g3 } is a subgroup of Aut( Gb f ).
2
5 3
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4
Figure 2. The butterﬂy graph (ﬁve nodes, with two node pairs connected by the bridging node 3). Table 2. The full automorphism group Aut( Gb f ) = {id, g1 , . . . , g7 } of the butterﬂy graph in Figure 2 and its effect on three partitions. Bold partitions are distinct. A possible generator is { g1 , g4 }. Permutation id = (1)(2)(3)(4)(5) g1 = ( 1 2 ) g2 = ( 4 5 ) g3 = (1 2)(4 5) g4 = (1 4)(2 5) g5 = (1 5)(2 4) g6 = ( 1 4 2 5 ) g7 = ( 1 5 2 4 )
P1 , Q = 0 {1, 2}, {3}, {4, 5} {2, 1}, {3}, {4, 5} {1, 2}, {3}, {5, 4} {2, 1}, {3}, {5, 4} {4, 5}, {3}, {1, 2} {5, 4}, {3}, {2, 1} {4, 5}, {3}, {2, 1} {5, 4}, {3}, {1, 2}
P2 , Q =
1 9
{1, 2, 3}, {4, 5} {2, 1, 3}, {4, 5} {1, 2, 3}, {5, 4} {2, 1, 3}, {5, 4} {4, 5, 3}, {1, 2} {5, 4, 3}, {2, 1} {4, 5, 3}, {2, 1} {5, 4, 3}, {1, 2}
1 P3 , Q = − 18
{1, 2, 3, 4}, {5} {2, 1, 3, 4}, {5} {1, 2, 3, 5}, {4} {2, 1, 3, 5}, {4} {4, 5, 3, 1}, {2} {5, 4, 3, 2}, {1} {4, 5, 3, 2}, {1} {5, 4, 3, 1}, {2}
Deﬁnition 2. Let G = (V, E) be a graph. A partition P is called stable, if P Aut(G)  = 1, otherwise it is called unstable. Stability here means that the automorphism group of the graph does not affect the given partition by tearing apart clusters.
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Example 2. Only P1 in Table 2 is stable because its orbit is trivial. The two modularity optimal partitions Aut( Gb f )
g
(e.g., P2id and P2 4 ) are not stable because P2
Aut( Gb f )
 = 2. Furthermore, P3
 = 4.
For the evaluation of graph clustering solutions, the effects of graph automorphisms on graph partitions are of considerable importance: 1. 2.
Automorphisms may lead to multiple equivalent optimal solutions as the butterﬂy graph shows g (P2id and P2 4 in Table 2). Partition comparison measures are not invariant with regard to automorphisms, as we show in Section 3.
3. Graph Partition Comparison Measures Are Not Invariant When comparing graph partitions, a natural requirement is that the partition comparison measure is invariant under automorphism. Deﬁnition 3. A partition comparison measure m : P(V ) × P(V ) → R is invariant under automorphism, if: ˜ m(P , Q) = m(P˜ , Q) for all P , Q ∈ P(V ) and P˜ ∈ P Aut(G) , Q˜ ∈ Q Aut(G) . Observe that if Q ∈ P Aut(G) , then such a measure m cannot distinguish between P and Q, since m(P , Q) = m(P , P ) by deﬁnition. However, unfortunately, as we show in the rest of this section, such a partition comparison measure does not exist. In the following, we present two proofs of this fact, which differ both in their level of generality and sophistication. 3.1. Variant 1: Construction of a Counterexample Theorem 2. The measures for comparing partitions deﬁned in Appendix B do not fulﬁll Deﬁnition 3 in general. Proof. We choose the cycle graph C36 and compute all modularity maximal partitions with Q = 2/3. Each of these six partitions has six clusters, and each of these clusters consists of a chain of six nodes (see Figure 3). Clearly, since all partitions are equivalent, an invariant partition comparison measure should identify them as equivalent: g0
g5
m(P0 , P0 ) = . . . = m(P0 , P0 )
(1)
gk m(P0 , P0 )
Computing for k = 0, . . . , 5 produces Table 3. Because the values in each row differ (in contrast to the requirements deﬁned by Equation (1)), each row of Table 3 contains the counterexample for the measure used. 3.2. Variant 2: Inconsistency of the Identity and the Invariance Axiom Theorem 3. Let G = (V, E) be a graph with V  > 2 and nontrivial Aut( G ). For partition comparison measures m : P(V ) × P(V ) → R, it is impossible to fulﬁll jointly the identity axiom m(P , Q) = c, if and only if P = Q (e.g., for a distance measure c = 0, for a similarity measure c = 1, etc.) for all P , Q ∈ P(V ) and the axiom of invariance (from Deﬁnition 3) m(P , Q) = c, ∀Q ∈ P Aut(G) . Proof. 1. 2.
Since Aut( G ) is nontrivial, a nontrivial orbit with at least two different partitions, namely P and Q, exists because P Aut(G)  > 1. It follows from the invariance axiom that m(P , Q) = c. The identity axiom implies that it follows from m(P , Q) = c that P = Q. 271
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3.
This contradicts the assumption that P and Q are different.
Table 3. Comparing the modularity maximizing partitions of the cycle graph C36 with modularity Q = 23 . The six optimal partitions consist of six clusters (see Figure 3). The number of pairs in the same cluster in both partitions is denoted by N11 , in different clusters by N00 and in the same cluster in one partition, but not in the other, by N01 or N10 . For the deﬁnitions of all partition comparison measures, see Appendix B. To compute this table, the R package partitionComparison has been used [23]. gk
m(P0 , P0 ) with g = (1 2 3 . . . 35 36) for k:
Measure 0
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Pair counting measures ( f ( N11 , N00 , N01 , N10 ); see Tables A1 and A2) RI ARI H CZ K MC P WI WII FM Γ SS1 B1 GL SS2 SS3 RT GK J RV RR M Mi Pe B2 LI NLI FMG
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.14286 0.0 0.0 0.00002 0.12245 24.37212 1.0 0.94730
LA dCE D
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MI NMI (max) NMI (min) NMI (Σ) VI
1.79176 1.0 1.0 1.0 0.0
0.90476 0.61111 0.80952 0.66667 0.66667 0.33333 0.61111 0.66667 0.66667 0.66667 0.61111 0.80556 0.91383 0.95 0.33333 0.62963 0.82609 0.94286 0.5 0.61039 0.09524 120.0 0.81650 0.00001 0.07483 14.89407 0.61111 0.61396
0.84762 0.37778 0.69524 0.46667 0.46667 −0.06667 0.37778 0.46667 0.46667 0.46667 0.37778 0.68889 0.87084 0.91753 0.17949 0.42519 0.73554 0.79937 0.30435 0.37662 0.06667 192.0 1.03280 0.00001 0.04626 9.20724 0.37778 0.41396
0.82857 0.3 0.65714 0.4 0.4 −0.2 0.3 0.4 0.4 0.4 0.3 0.65 0.85796 0.90625 0.14286 0.36 0.70732 0.71429 0.25 0.29870 0.05714 216.0 1.09545 0.00001 0.03673 7.31163 0.3 0.34730
0.84762 0.37778 0.69524 0.46667 0.46667 −0.06667 0.37778 0.46667 0.46667 0.46667 0.37778 0.68889 0.87084 0.91753 0.17949 0.42519 0.73554 0.79937 0.30435 0.37662 0.06667 192.0 1.03280 0.00001 0.04626 9.20724 0.37778 0.41396
0.90476 0.61111 0.80952 0.66667 0.66667 0.33333 0.61111 0.66667 0.66667 0.66667 0.61111 0.80556 0.91383 0.95 0.33333 0.62963 0.82609 0.94286 0.5 0.61039 0.09524 120.0 0.81650 0.00001 0.07483 14.89407 0.61111 0.61396
Setbased comparison measures (see Table A3) 0.83333 0.16667 12.0
0.66667 0.33333 24.0
0.5 0.5 36.0
0.66667 0.33333 24.0
0.83333 0.16667 12.0
Information theorybased measures (see Table A4) 1.34120 0.74854 0.74854 0.74854 0.90112
1.15525 0.64475 0.64475 0.64475 1.27303
272
1.09861 0.61315 0.61315 0.61315 1.38629
1.15525 0.64475 0.64475 0.64475 1.27303
1.34120 0.74854 0.74854 0.74854 0.90112
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Figure 3. The cycle graph C36 (the “outer” cycle) and an initial partition of six clusters (connected nodes of the same color, separated by dashed lines). A single application of g = (1 2 . . . 36) “rotates” the g graph by one node (the “inner” cycle C36 ). As a consequence, in each cluster, one node drops out and is added to another cluster: For instance, Node 1 drops out of the “original” cluster C = {1, 2, 3, 4, 5, 6}, and Node 7 is added, resulting in C g = {2, 3, 4, 5, 6, 7}. All dropped nodes are shown in light gray.
4. The Construction of Invariant Measures for Finite Permutation Groups The purpose of this section is to construct invariant counterparts for most of the partition comparison measures in Appendix B. We proceed in two steps: 1. 2.
We construct a pseudometric space from the images of the actions of Aut( G ) on partitions in P(V ) (Deﬁnition 1). We extend the metrics for partition comparison by constructing invariant metrics on the pseudometric space of partitions.
4.1. The Construction of the Pseudometric Space of Equivalence Classes of Graph Partitions We use a variant of the idea of Doob’s concept of a pseudometric space [24] (p. 5). A metric for a space S (with s, t, u ∈ S) is a function d : S × S → R+ for which the following holds: 1.
Symmetry: d(s, t) = d(t, s). 273
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2. 3.
Identity: d(s, t) = 0 if and only if s = t. Triangle inequality: d(s, u) ≤ d(s, t) + d(t, u).
A pseudometric space (S, d∗ ) relaxes the identity condition to d∗ (s, s) = 0. The distance between two elements s1 , s2 of an equivalence class [s] is deﬁned as d∗ (s1 , s2 ) = 0 by Deﬁnition 3. For graphs, S is the finite set of partitions P(V ) and S∗ is the partition of P(V ) into orbits of Aut(G): # $ S∗ (V ) = P(V ) Aut(G) = P Aut(G)  P ∈ P(V ) . A partition P in S corresponds to its orbit P Aut(G) in S∗ . The relations between the spaces used in the following are: 1. 2. 3.
(S, d) is a metric space with S = P(V ) and with the function d : P(V ) × P(V ) → R. (S∗ , d∗ ) is a metric space with S∗ = P(V ) Aut(G) = {P Aut(G)  P ∈ P(V )} and the function d∗ : P(V ) Aut(G) × P(V ) Aut(G) → R. We construct three variants of d∗ in Section 4.2. (S, d∗ ) is the pseudometric space with S = P(V ) and with the metric d∗ . The partitions in S are mapped to arguments of d∗ by the transformation ec : P(V ) → P(V ) Aut(G) , which is deﬁned as ec(P ) := P Aut(G) .
Table 4 illustrates S∗ (the space of equivalence classes) of the pseudometric space (S, d∗ ) of the butterﬂy graph (shown in Figure 2). S∗ is the partition of P({1, 2, 3, 4, 5}) into 17 equivalence classes. Only the four classes E1 , E8 , E12 and E17 are stable because they are trivial orbits. The three partitions from Table 2 are contained in the following equivalence classes: P1 ∈ E8 , P2 ∈ E14 , and P3 ∈ E13 . Table 4. The equivalence classes of the pseudometric space (S, d∗ ) of the butterﬂy graph (see Figure 2). Classes are grouped by their partition type, which is the corresponding integer partition. k is the number of partitions per type; l is the number of clusters the partitions of a type consists of; dia1− RI is the diameter (see Equation (2)) of the equivalence class computed for the distance d RI computed from the Rand Index (RI) by 1 − RI.
P Aut (G)
Q
dia1− RI
− 29
0.0
− 19 − 16
0.2 0.2
5 − 18
0.2
0 − 29
0.6 0.4
− 16
0.6
0 − 13 1 − 18
0.0 0.4 0.4
− 29
0.4
Partition type (1, 1, 1, 1, 1), k = 1, l = 5 E1
{1}, {2}, {3}, {4}, {5} Partition type (1, 1, 1, 2), k = 10, l = 4
E2 E3 E4
{1}, {2}, {3}, {4, 5} {1, 2}, {3}, {4}, {5} {1}, {2}, {3, 4}, {5} {1}, {2}, {3, 5}, {4} {1}, {2, 3}, {4}, {5} {1, 3}, {2}, {4}, {5} {1}, {2, 4}, {3}, {5} {1}, {2, 5}, {3}, {4} {1, 4}, {2}, {3}, {5} {1, 5}, {2}, {3}, {4} Partition type (1, 1, 3) k = 10, l = 3
E5 E6 E7
{1}, {2}, {3, 4, 5} {4}, {5}, {1, 2, 3} {1}, {3}, {2, 4, 5} {3}, {5}, {1, 2, 4} {3}, {4}, {1, 2, 5} {2}, {3}, {1, 4, 5} {1}, {5}, {2, 3, 4} {1}, {4}, {2, 3, 5} {2}, {5}, {1, 3, 4} {2}, {4}, {1, 3, 5} Partition type (1, 2, 2), k = 15, l = 3
E8 E9 E10 E11
{3}, {1, 2}, {4, 5} {3}, {1, 4}, {2, 5} {1}, {2, 3}, {4, 5} {2}, {1, 3}, {4, 5} {1}, {2, 4}, {3, 5} {4}, {1, 3}, {2, 5} {2}, {1, 5}, {3, 4}
{3}, {1, 5}, {2, 4} {5}, {1, 2}, {3, 4} {4}, {1, 2}, {3, 5} {1}, {2, 5}, {3, 4} {5}, {1, 3}, {2, 4} {2}, {1, 4}, {3, 5} {5}, {1, 4}, {2, 3} {4}, {1, 5}, {2, 3}
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Table 4. Cont.
P Aut (G)
Q
dia1− RI
− 29 1 − 18
0.0 0.6
− 16
1 9
0.4 0.6
− 29
0.6
0
0.0
Partition type (1, 4), k = 5, l = 2 E12 E13
{1, 2, 4, 5}, {3} {2, 3, 4, 5}, {1} {1, 2, 3, 4}, {5} {1, 2, 3, 5}, {4} {1, 3, 4, 5}, {2} Partition type (2, 3), k = 10, l = 2
E14 E15 E16
{1, 2}, {3, 4, 5} {4, 5}, {1, 2, 3} {3, 5}, {1, 2, 4} {3, 4}, {1, 2, 5} {1, 3}, {2, 4, 5} {2, 3}, {1, 4, 5} {2, 5}, {1, 3, 4} {2, 4}, {1, 3, 5} {1, 4}, {2, 3, 5} {1, 5}, {2, 3, 4} Partition type (5), k = 1, l = 1
E17
{1, 2, 3, 4, 5}
4.2. The Construction of LeftInvariant and Additive Measures on the Pseudometric Space of Equivalence Classes of Graph Partitions In the following, we consider only partition comparison measures, which are distance functions of a metric space. Note that a normalized similarity measure s can be transformed into a distance by the transformation d = 1 − s. In a pseudometric space (S, d∗ ), we measure the distance d∗ (P , Q) between equivalence classes (which are sets) of partitions instead of the distance d(P , Q) between partitions. The partitions P and Q are formal arguments of d∗ , which are expanded to equivalence classes by P Aut(G) and Q Aut(G) . The standard construction of a distance measure between sets has been developed for the point set topology and is due to Felix Hausdorff [25] (p. 166) and Kazimierz Kuratowski [26] (p. 209). For ﬁnite sets, it requires the computation of the distances for all pairs of the direct product of the two sets. Since for ﬁnite permutation groups, we deal with distances between two ﬁnite sets of partitions, we use the following deﬁnitions for the lower and upper measures, respectively. Both deﬁnitions have the form of an optimization problem: ˜ d∗ (P , Q) = min d(P˜ , Q) L
and: ∗ (P , Q) dU
=
P˜ ∈P Aut(G) , Aut( G ) ˜ Q∈Q
⎧ ⎨
0 ˜ max d(P˜ , Q) Aut ˜ P ∈P (G) , ⎩
if P Aut(G) = Q Aut(G) else
Aut( G ) ˜ Q∈Q
The diameter of a ﬁnite equivalence class of partitions is deﬁned by dia(P ) =
max
P˜ ∈P Aut(G) , Aut( G ) ˜ Q∈P
˜ . d(P˜ , Q)
(2)
The third option of deﬁning a distance between two ﬁnite equivalence classes of partitions of taking the average distance is due to John von Neumann [27]: d∗av (P , Q)
=
⎧ ⎨ ⎩
1 P Aut(G) ·Q Aut(G) 
0 ˜ ∑P˜ ∈P Aut(G) , d(P˜ , Q) Aut( G ) ˜ Q∈Q
275
if P Aut(G) = Q Aut(G) else
Symmetry 2018, 10, 504
∗ and d∗ require the computation of the minimal, maximal and Note that the deﬁnitions for d∗L , dU av average distance of all pairs of the direct product P Aut(G) × Q Aut(G) . The computational complexity of this is quadratic in the size of the larger equivalence class. Posed as a measurement problem, we can instead ﬁx one partition in one of the orbits and measure the minimal, maximal and average distance between all pairs of either the direct product of {P } × Q Aut(G) or {Q} × P Aut(G) . The complexity of this is linear in the size of the smaller equivalence class. Theorems 4 and 5 and their proofs are based on these observations. They are the basis for the development of algorithms for the computation of invariant partition comparison measures of a computational complexity of at most linear order and often of constant order.
Theorem 4. For all P Aut(G) = Q Aut(G) , the following equations hold: d∗L (P , Q) =
= =
min
˜ = d(P˜ , Q)
min
˜ = d(P , Q)
min
d(P˜ , Q) =
max
˜ = d(P˜ , Q)
max
˜ = d(P , Q)
max
d(P˜ , Q) =
P˜ ∈P Aut(G) , Aut( G ) ˜ Q∈Q ˜ Q Aut(G) Q∈
P˜ ∈ P Aut(G)
min
g,h∈ Aut( G )
d(P h , Q g )
min
d(P , Q g )
min
d(P h , Q)
g∈ Aut( G ) h∈ Aut( G )
For P Aut(G) = Q Aut(G) : ∗ (P , Q) = dU
= =
P˜ ∈P Aut(G) , Aut( G ) ˜ Q∈P ˜ Q Aut(G) Q∈
P˜ ∈ P Aut(G)
max
g,h∈ Aut( G )
d(P h , Q g )
max d(P , Q g )
g∈ Aut( G )
max d(P h , Q)
h∈ Aut( G )
Proof. Let g, h, f ∈ Aut( G ), P˜ ∈ P Aut(G) and Q˜ ∈ Q Aut(G) , that is P˜ = P h and Q˜ = Q g . Then, since the orbits of both partitions are generated by Aut( G ), the following identities between distances hold: −1
˜ = d(P , Q g ) = d(P g , Q), d(P , Q) −1
d(P˜ , Q) = d(P h , Q) = d(P , Qh ) as well as:
˜ = d(P h , Q g ) = d(P hg−1 , Q), d(P˜ , Q)
and:
˜ = d(P h , Q g ) = d(P , Q gh−1 ). d(P˜ , Q) Furthermore, let f = gh−1 .
1.
For d∗L , we have: min
Aut( G ) ˜ Q∈Q
˜ = d(P , Q)
=
min
g∈ Aut( G )
d(P , Q g )
min
g−1 ∈ Aut( G )
−1
d(P g , Q) =
min
P˜ ∈P Aut(G)
d(P˜ , Q)
by switching the reference systems. In the next sequence of equations, we establish that taking the minimum over all reference systems is equivalent to ﬁnding the minimum for one arbitrarily ﬁxed reference system.
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Symmetry 2018, 10, 504
min
P˜ ∈P Aut(G) , Aut( G ) ˜ Q∈Q
˜ = d(P˜ , Q)
= 2.
min
g,h∈ Aut( G )
min
f ∈ Aut( G )
d(P h , Q g ) =
d(P , Q f )
=
−1
min
d(P , Q gh )
min
˜ d(P , Q)
g,h∈ Aut( G )
Aut( G ) ˜ Q∈Q
∗ for P Aut( G ) = Q Aut( G ) we substitute max for min in the proof of d∗ . For the proof of dU L
Theorem 5. For all P Aut(G) = Q Aut(G) , the following equations hold: d∗av (P , Q) =
1 ˜ ∑ d(P˜ , Q) P Aut(G)  · Q Aut(G)  P˜ ∈P Aut(G) ,
(3)
Aut( G ) ˜ Q∈Q
1 =  Aut( G )2
∑
d(P h , Q g )
(4)
h,g∈ Aut( G )
=
1 ∑ d(P˜ , Q) P Aut(G)  P˜ ∈P Aut(G)
(5)
=
1  Aut( G )
d(P h , Q)
(6)
=
1 ˜ ∑ d(P , Q) Q Aut(G)  Q∈Q Aut( G ) ˜
(7)
=
1  Aut( G )
(8)
∑
h∈ Aut( G )
∑
d(P , Q g )
g∈ Aut( G )
Proof. For the proof of the equality of the identities of d∗av , we use the property of an average of n observations xi,j with k identical groups of size m with i ∈ 1, . . . , k, j ∈ 1, . . . , m: 1 n
k
m
∑ ∑ xi,j =
i =1 j =1
k km
m
∑ x1,j =
j =1
1 m
m
∑ x1,j
(9)
j =1
The computation of an average over the group equals the result of the computation of an average over the orbit, because the orbit stabilizer Theorem 1 implies that each element of the orbit is generated  Aut( G )P  times, and this means that we average  Aut( G )P  groups of identical values and that Equation (9) applies. This establishes the equality of Expressions (3) and (4), as well as of Expressions (5) and (6) and of Expressions (7) and (8), respectively. The two decompositions of the direct product Aut( G ) × Aut( G ) establish the equality of Expressions (4) and (6), as well as of Expressions (4) and (8). ∗ (P , Q) and d∗ (P , Q) are invariant. Next, we Note that these proofs also show that d∗L (P , Q), dU av ∗ (P , Q) and d∗ (P , Q) are invariant measures. prove that the three measures d∗L (P , Q), dU av
Theorem 6. The lower pseudometric space (S, d∗L ) has the following properties: 1. 2. 3. 4.
Identity: d∗L (P , Q) = 0, if P Aut(G) = Q Aut(G) . ˜ , for all P , Q ∈ P(V ) and P˜ ∈ P Aut(G) , Q˜ ∈ Q Aut(G) . Invariance: d∗L (P , Q) = d∗L (P˜ , Q) Symmetry: d∗L (P , Q) = d∗L (Q, P ). Triangle inequality: d∗L (P , R) ≤ d∗L (P , Q) + d∗L (Q, R)
∗ ) and the average pseudometric These properties also hold for the upper pseudometric space (S, dU space (S, d∗av ).
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Proof. 1. 2. 3. 4.
Identity holds because of the deﬁnition of the distance d∗ between two elements in an equivalence class of the pseudometric space (S, d∗ ). ∗ (P , Q) and d∗ (P , Q) is proven by Theorems 4 and 5. Invariance of d∗L (P , Q), dU av Symmetry holds, because d is symmetric, and min, max and the average do not depend on the order of their respective arguments. To proof the triangular inequality, we make use of Theorems 4 and 5 and of the fact that d is a metric for which the triangular inequality holds: (a)
For d∗L follows: d∗L (P , R) =
≤
= = =
(b) (c)
min
P˜ ∈P Aut(G) , Aut( G ) ˜ R∈R
min
P˜ ∈P Aut(G) , Aut( G ) , ˜ Q∈Q Aut( G ) ˜ R∈R
min
P˜ ∈P Aut(G) , Aut( G ) ˜ R∈R
˜ d(P˜ , R)
˜ + d(Q˜ , R) ˜ d(P˜ , Q)
˜ d(P˜ , Q) + d(Q, R)
d(P˜ , Q) +
min
min
Aut( G ) ˜ P˜ ∈P Aut(G) R∈R d∗L (P , Q) + d∗L (Q, R)
˜ d(Q, R)
∗ , we substitute max for min and d for d in For the proof of the triangular inequality for dU U L ∗ the proof of the triangular inequality for d L . For d∗av , it follows:
1 ˜ ∑ ∑ d(P˜ , R) P Aut(G)  · R Aut(G)  P˜ ∈P Aut(G) R∈R Aut( G ) ˜ ; < 1 ˜ ≤ ∑ ∑ d(P˜ , Q) + d(Q, R) P Aut(G)  · R Aut(G)  P˜ R˜
d∗av (P , R) =
=
1 1 ˜ ∑ ∑ d(P˜ , Q) + P Aut(G)  · R Aut(G)  ∑ ∑ d(Q, R) P Aut(G)  · R Aut(G)  P˜ R˜ ˜ P˜ R
=
1 1 ∑ d∗av (P , Q) + P Aut(G)  ∑ d∗av (Q, R) R Aut(G)  R˜ P˜
= d∗av (P , Q) + d(Q˜ , R) 5. Decomposition of Partition Comparison Measures In this section, we assess the structural (dis)similarity between two partitions and the effect of the group actions by combining a partition comparison measure and its invariant counterpart deﬁned ∗ (P , Q) and d∗ (P , Q) allow the decomposition of a in Section 4. The distances d(P , Q), d∗L (P , Q), dU av partition comparison measure (transformed into a distance) into a structural component dstruc and the effect d Aut(G) of the automorphism group Aut( G ):
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Symmetry 2018, 10, 504
d(P , Q) = d∗L (P , Q) + (d(P , Q) − d∗L (P , Q)) 9 :7 8 9 :7 8 dstruc
=
d Aut(G)
∗ ∗ dU (P , Q) − (dU (P , Q) − d(P , Q))
9
:7
8
9
:7
dstruc
8
d Aut(G)
= d∗av (P , Q) − (d∗av (P , Q) − d(P , Q)) 9 :7 8 9 :7 8 dstruc
d Aut(G)
dia(P ) measures the effect of the automorphism group Aut( G ) on the equivalence class P Aut(G) Aut( G )
(see the last column of Table 4). emax of two partitions P and Q:
is an upper bound of the automorphism effect on the distance
Aut( G )
emax
= min(dia(P ), dia(Q)). Aut( G )
This follows from Theorem 4. Note that emax
∗ − d∗ , as Case 1 in Table 5 shows. ≥ dU L
Table 5. Measure decomposition for partitions of the butterﬂy graph for the Rand distance d RI = 1 − RI. Case
P
Q
d RI
d∗
dstruc
d Aut (G)
d∗L d∗av ∗ dU d∗L d∗av ∗ dU d∗L d∗av ∗ dU d∗L d∗av ∗ dU
0.3 0.5 0.7
0.0 0.2 0.4
0.2 0.4 0.6
0.4 0.2 0.0
0.1 0.25 0.3
0.2 0.05 0.0
0.3 0.3 0.3
0.0 0.0 0.0
1
{{1, 2, 3, 4}{5}} ∈ E13 dia( E13 ) = 0.6
{{4}{5}{1, 2, 3}} ∈ E5 dia( E5 ) = 0.6
0.3 0.3 0.3
2
{{2, 4}{1, 3, 5}} ∈ E16 dia( E16 ) = 0.6
{{3}{1, 4}{2, 5}} ∈ E9 dia( E9 ) = 0.4
0.6 0.6 0.6
3
{{1}{2, 5}{3, 4}} ∈ E11 dia( E11 ) = 0.4
{{1}{2, 3}{4}{5}} ∈ E3 dia( E3 ) = 0.2
0.3 0.3 0.3
4
{{3}{1, 2}{4, 5}} ∈ E8 (dia( E8 ) = 0, stable)
{{1}{2, 3}{4, 5}} ∈ E10 dia( E10 ) = 0.4
0.3 0.3 0.3
In Table 5, we show a few examples of measure decomposition for partitions of the butterﬂy graph for the Rand distance d RI : 1.
2. 3.
4.
In Case 1, we compare two partitions from nontrivial equivalence classes: the difference of 0.4 ∗ and d∗ indicates that the potential maximal automorphism effect is larger than the between dU L lower measure. In addition, it is also smaller (by 0.2) than the automorphism effect in each of the equivalence classes. That d Aut(G) is zero for the lower measure implies that the pair (P , Q) is a pair with the minimal distance between the equivalence classes. The fact that d∗av = 0.5 is the midpoint between the lower and upper measures indicates a symmetric distribution of the distances between the equivalence classes. That d Aut(G) is zero for the upper measure in Case 2 means that we have found a pair with the maximal distance between the equivalence classes. In Case 3, we have also found a pair with maximal distance between the equivalence classes. However, the maximal potential automorphism effect is smaller than for Cases 1 and 2. In addition, the distribution of distances between the equivalence classes is asymmetric. Case 4 shows the comparison of a partition from a trivial with a partition from a nontrivial equivalence class. Note, that in this case, all three invariant measures, as well as d RI coincide and that no automorphism effect exists.
A different approach to measure the potential instability in clustering a graph G is the computation of the Kolmogorov–Sinai entropy of the ﬁnite permutation group Aut( G ) acting on the graph [28]. 279
Symmetry 2018, 10, 504
Note, that the Kolmogorov–Sinai entropy of a ﬁnite permutation group is a measure of the uncertainty of the automorphism group. It cannot be used as a measure to compare two graph partitions. 6. Invariant Measures for the Karate Graph In this section, we illustrate the use of invariant measures for the three partitions PO , P1 and P2 of the Karate graph K [29], which is shown in Figure 4. Aut(K ) is of order 480, and it consists of the three subgroups G1 = Sym(Ω1 ) with Ω1 = {15, 16, 19, 21, 23}, G2 = Sym(Ω2 ) with Ω2 = {18, 22} and G3 = {(), (5 11), (6 7)}. In addition to the modularity optimal partition PO (with its clusters separated by longer and dashed lines in Figure 4), we use the partitions P1 and P2 :
P1 = {{5, 6, 7, 19, 21} , {1, 2, 3, 4, 8, 12, 13, 14, 18, 20, 22} , {9, 10, 11, 15, 16, 17, 23, 27, 30, 31, 33, 34} , {24, 25, 26, 28, 29, 32}} P2 = {{5, 6, 7, 8, 12, 19, 21} , {1, 2, 3, 4, 13, 14, 18, 20, 22} , {9, 10, 11, 15, 16, 17, 23, 27, 30, 31, 33, 34} , {24, 25, 26, 28, 29, 32}} Both partitions are affected by the orbits {15, 16, 19, 21, 23} and {5, 11}, each overlapping two clusters. The dissimilarity to PO is larger for P2 , which is reﬂected in Tables 6 and 7. 23
21
C3
17 6 7
31
C1
19 16
33
15 9
C2
11 5
22
27
18
30
34
20 10
2
14
1 3 4
12
28
8 29
13
25 32
24 C4 26
Figure 4. Zachary’s Karate graph K with the vertices of the orbits of the three subgroups of Aut(K ) in bold and the clusters of PO separated by dashed edges.
For the optimal partition PO of type (5, 6, 11, 12), the upper bound of the size of the equivalence class is 480 [30] (p. 112). The actual size of the equivalence class of PO is one, which means the optimal solution is not affected by Aut(K ). Partition P1 , which is of the same type as PO , also has an upper bound of 480 for its equivalence class. The actual size of the equivalence classes of both P1 and P2 is 20. Note that the actual size of the equivalence classes that drive the complexity of computing invariant measures is in our example far below the upper bound. Table 6 shows the diameters of the equivalence classes of the partitions.
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Symmetry 2018, 10, 504
Table 6. Diameter (computed using d RI ), orbit size and stability of partitions PO , P1 and P2 .
X
PO
P1
P2
dia(X ) X Aut(G)  X stable?
0.0000 1 yes
0.1176 20 no
0.1390 20 no
Table 7 illustrates the decomposition into structural effects and automorphism effects for the three partitions of the Karate graph. We see that for the comparison of a stable partition (PO ) with one of the unstable partitions, the classic partition comparison measures are sufﬁcient. However, when comparing the two unstable partitions P1 and P2 , the structural effect (0.0499) is dominated by the maximal automorphism effect (0.1176). Furthermore, we note that the distribution of values over the ∗ and d∗ ). orbit of the automorphism group is asymmetric (by looking at d∗L , dU av Table 7. Invariant measures and automorphism effects for the Karate graph. The R package partitionComparison has been used for the computations [23]. Measure d = d RI
m(PO , P1 )
m(PO , P2 )
m(P1 , P2 )
d d∗L + d Aut(G) ∗ −d dU Aut( G ) d∗av − d Aut(G)
0.0927 0.0927 0.0927 0.0927
0.1426 0.1426 0.1426 0.1426
0.0499 0.0499 + 0.0000 0.1676 − 0.1176 0.1280 − 0.0781
0.0000
0.0000
0.1176
Aut(K )
emax
The analysis of the effects of the automorphism group of the Karate network showed that the automorphism group does not affect the stability of the optimal partition. However, the ﬁrst results show that the situation is different for other networks like the Internet AS graph with 40,164 nodes and 85,123 edges (see Rossi et al. [31], and the data of of the graph techinternetas are from Rossi and Ahmed [32]): for this graph, several locally optimal solutions with a modularity value above 0.694 exist, all of which are unstable. Further analysis of the structural properties of the solution landscape of this graph is work in progress. 7. Discussion, Conclusions and Outlook In this contribution, we study the effects of graph automorphisms on partition comparison measures. Our main results are: 1. 2. 3. 4. 5. 6.
A formal deﬁnition of partition stability, namely P is stable iff P Aut(G)  = 1. A proof of the noninvariance of all partition comparison measures if the automorphism group is nontrivial ( Aut( G ) > 1). The construction of a pseudometric space of equivalence classes of graph partitions for three classes of invariant measures concerning ﬁnite permutation groups of graph automorphisms. The proof that the measures are invariant and that for these measures (after the transformation to a distance), the axioms of a metric space hold. The space of partitions is equipped with a metric (the original partition comparison measure) and a pseudometric (the invariant partition comparison measure). The decomposition of the value of a partition comparison measure into a structural part and a remainder that measures the effect of group actions.
Our deﬁnitions of invariant measures have the advantage that any existing partition comparison measure (as long as it is a distance or can be transformed into one) can still be used for the task. Moreover, the decomposition of measures restores the primary purpose of the existing comparison 281
Symmetry 2018, 10, 504
measures, which is to quantify structural difference. However, the construction of these measures leads directly to the classic graph isomorphism problem, whose complexity—despite considerable efforts and hopes to the contrary [33]—is still an open theoretical problem [34,35]. However, from a pragmatic point of view, today, quite efﬁcient and practically usable algorithms exist to tackle the graph isomorphism problem [34]. In addition, for very large and sparse graphs, algorithms for ﬁnding generators of the automorphism group exist [11]. Therefore, this dependence on a computationally hard problem in general is not an actual disadvantage and allows one to implement the presented measure decomposition. The efﬁcient implementation of algorithms for the decomposition of graph partition comparison measures is left for further research. Another constraint is that we have investigated the effects of automorphisms on partition comparison measures in the setting of graph clustering only. The reason for this restriction is that the automorphism group of the graph is already deﬁned by the graph itself and, therefore, is completely contained in the graph data. For arbitrary datasets, the information about the automorphism group is usually not contained in the data, but must be inferred from background theories. However, provided we know the automorphism group, our results on the decomposition of the measures generalize to arbitrary cluster problems. All in all, this means that this article provides two major assets: ﬁrst, it provides a theoretic framework that is independent of the preferred measure and the data. Second, we provide insights into a source of possible partition instability that has not yet been discussed in the literature. The downsides (symmetry group must be known and graph clustering only) are in our opinion not too severe, as we discussed above. Therefore, we think that our study indicates that a better understanding of the principle of symmetry is important for future research in data analysis. Supplementary Materials: The R package partitionComparison by the authors of this article that implements the different partition comparison measures is available at https://cran.rproject.org/package=partitionComparison. Author Contributions: The F.B. implemented the R package mentioned in the Supplementary Materials and conducted the noninvariance proof by counterexample. The more general proof of the nonexistence of invariant measures, as well as the idea of creating a pseudometric to repair a measure’s deﬁciency of not being invariant is mainly due to the A.G.S. Both authors contributed equally to writing the article and revising it multiple times. Funding: We acknowledge support by Deutsche Forschungsgemeinschaft and the Open Access Publishing Fund of the Karlsruhe Institute of Technology. Acknowledgments: We thank Andreas GeyerSchulz (Institute of Analysis, Faculty of Mathematics, KIT, Karlsruhe) for repeated corrections and suggestions for improvement in the proofs. Conﬂicts of Interest: The authors declare no conﬂict of interest.
Appendix A. Modularity Newman’s and Girvan’s modularity [14] is deﬁned as: Q=
∑
eii − a2i
i
with the edge fractions: eij =
& & &{{u, v} ∈ Eu ∈ Ci ∧ v ∈ Cj }& , i = j, 2 E 
and the cluster density: eii =
{{u, v} ∈ Eu, v ∈ Ci } .  E
We have to distinguish eij and eii because of the setbased deﬁnition E. eij is the fraction of edges from cluster Ci to cluster Cj and e ji , vice versa. Therefore, the edges are counted twice, and thus, the fraction has to be weighted with 12 . The second part of Q is the marginal distribution:
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Symmetry 2018, 10, 504
a2i
=
2
∑ eij
.
j
High values of Q indicate good partitions. The range of Q is [− 12 , 1). Even if the modularity has some problems by design (e.g., the resolution limit [36], unbalanced cluster sizes [37], multiple equivalent, but unstable solutions generated by automorphisms [38]), maximization of Q is the de facto standard formal optimization criterion for graph clustering algorithms. Appendix B. Measures for Comparing Partitions We classify the measures that are used in the literature to compare object partitions as three categories [39]: 1. 2. 3.
Paircounting measures. Setbased comparison measures. Information theory based measures.
All these measures come from a general context and, therefore, may be used to compare any object partitions, not only graph partitions. The ﬂip side of the coin is that they do not consider any adjacency information from the underlying graph at all. The column Abbr. of Tables A1–A4 denotes the Abbreviations used throughout this paper; the column P = P denotes the value resulting when identical partitions are compared (max stands for some maximum value depending on the partition). Appendix B.1. PairCounting Measures All the measures within the ﬁrst class are based on the four coefﬁcients Nxy that count pairs of objects (nodes in our context). Let P , Q be partitions of the node set V of a graph G. C and C denote clusters (subsets of vertices C, C ⊆ V). The coefﬁcients are deﬁned as: &3 N11 := & {u, v} &3 N10 := & {u, v} &3 N01 := & {u, v} &3 N00 := & {u, v}
4& ⊆ V  (∃C ∈ P : {u, v} ⊆ C ) ∧ (∃C ∈ Q : {u, v} ⊆ C ) & , 4& ⊆ V  (∃C ∈ P : {u, v} ⊆ C ) ∧ (∀C ∈ Q : {u, v} ⊆ C ) & , 4& ⊆ V  (∀C ∈ P : {u, v} ⊆ C ) ∧ (∃C ∈ Q : {u, v} ⊆ C ) & , 4& ⊆ V  (∀C ∈ P : {u, v} ⊆ C ) ∧ (∀C ∈ Q : {u, v} ⊆ C ) & . n ( n −1)
Please note that N11 + N10 + N01 + N00 = (n2 ) = . One easily can see that for identical 2 partitions N10 = N01 = 0, because two nodes either occur in a cluster together or not. Completely different partitions result in N11 = 0. All the measures we examined are given in Tables A1 and A2. The RV coefﬁcient is used by Youness and Saporta [40] for partition comparison, and p and q are the cluster counts (e.g., p = P ) for the two partitions. For a detailed deﬁnition of the Lerman index (especially the deﬁnitions of the expectation and standard deviation), see Denœud and Guénoche [41].
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Table A1. The pair counting measures used in Table 3 [42]. The above measures are similarity measures. Distance measures and nonnormalized measures are listed in Table A2. For brevity: N21 = N11 + N10 , = N + N and N = N + N . Abbr., Abbreviation. N12 = N11 + N01 , N01 00 00 01 10 10 Abbr.
Measure
Formula
P=P
RI
Rand [43]
N11 + N00 (n2 )
1.0
ARI
Hubert and Arabie [44]
2( N00 N11 − N10 N01 ) N +N N N01 12 10 21
1.0
H
Hamann [45]
( N11 + N00 )−( N10 + N01 ) (n2 )
1.0
Czekanowski [46]
2N11 2N11 + N10 + N01
1.0
CZ
1 2
N11 N21
+
N11 N12
1.0
K
Kulczynski [47]
MC
McConnaughey [48]
2 N11 − N10 N01 N21 N12
1.0
P
Peirce [49]
N11 N00 − N10 N01 N21 N01
1.0
WI
Wallace [50]
N11 N21
1.0
WII
Wallace [50]
N11 N12
1.0
FM
Fowlkes and Mallows [51]
N11 N11 N21 N12
1.0
Γ
Yule [52]
SS1
N11 N00 − N10 N01 √ 1 4
Sokal and Sneath [53]
1.0
N21 N12 N10 N01
N11 N21
+
N11 N12
+
N00 N10
+
N00 N01
1.0
B1
Baulieu [54]
2 (n2 ) −(n2 )( N10 + N01 )+( N10 − N01 )2 2 (n2 )
GL
Gower and Legendre [55]
N11 + N00 N11 + 12 ( N10 + N01 )+ N00
1.0
SS2
Sokal and Sneath [53]
N11 N11 +2( N10 + N01 )
1.0
SS3
Sokal and Sneath [53]
√
1.0
RT
Rogers and Tanimoto [56]
N11 + N00 N11 +2( N10 + N01 )+ N00
1.0
Goodman and Kruskal [57]
N11 N00 − N10 N01 N11 N00 + N10 N01
1.0
Jaccard [3]
N11 N11 + N10 + N01
GK J
RV
Robert and Escouﬁer [58]
N11 N00 N N21 N12 N01 10
1 N11 − 1q N21 − 1p N12 + (n2 ) pq 0 p −2 n 1 p N21 + ( 2 ) p2 1− 1 2 q −2 n 1 q N12 + ( 2 ) q2
284
1.0
1.0 1.0
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Table A2. Pair counting measures that are not similarity measures. For brevity: N21 = N11 + N10 , = N + N and N = N + N . N12 = N11 + N01 , N01 00 00 01 10 10 Abbr.
Measure
Formula
P=P
RR
Russel and Rao [59]
N11 (n2 )
max
M
Mirkin and Chernyi [60]
2( N01 + N10 )
0.0
Mi
N10 + N01 N11 + N10
Hilbert [61]
0.0
Pe
Pearson [62]
N11 N00 − N10 N01 N N21 N12 N01 10
B2
Baulieu [54]
N11 N00 − N10 N01 2 (n2 )
max
LI
Lerman [63]
N11 − E( N11 ) √ 2 σ ( N11 )
max
NLI
Lerman [63] (normalized)
LI( P1 ,P2 ) LI( P1 ,P1 )LI( P2 ,P2 )
1.0
FMG
√ N11 N21 N12
Fager and McGowan [64]
−
√1 2 N21
max
max
Appendix B.2. SetBased Comparison Measures The second class is based on plain set comparison. We investigate three measures (see Table A3), namely the measure of Larsen and Aone [65], the socalled classiﬁcation error distance [66] and Dongen’s metric [67]. Table A3. References and formulas for the three setbased comparison measures used in Table 3. σ is the result of a maximum weighted matching of a bipartite graph. The bipartite graph is constructed from the partitions that shall be compared: the two node sets are derived from the two partitions, and each cluster is represented by a node. &By deﬁnition, the two node& sets are disjoint. The node sets are & & connected by edges of weight wij = &{Ci ∩ Cj  Ci ∈ P , Cj ∈ Q}&. As in our context P  = Q, the found σ is assured to be a perfect (bijective) matching. n is the number of nodes V . Abbr.
Measure
LA
Larsen and Aone [65]
dCE
Meilˇa and Heckerman [66]
D
P=P
Formula
van Dongen [67]
1 P 
1−
2 C ∩ C  ∑C∈P maxC ∈Q C+C  1 n
maxσ ∑C∈P C ∩ σ(C )
C ∩ C −
2n − ∑C∈P maxC ∈Q ∑C ∈Q maxC∈P C ∩ C 
1.0 0.0 0.0
Appendix B.3. Information TheoryBased Measures The last class of measures contains those that are rooted in information theory. We show the measures in Table A4, and we recap the fundamentals brieﬂy: the entropy of a random variable X is deﬁned as: k
H ( X ) = − ∑ pi log pi i =1
with pi being the probability of a speciﬁc incidence. The entropy of a partition can analogously be deﬁned as: C  C  H (P ) = − ∑ log . n n C ∈P
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The mutual information of two random variables is: I ( X, Y ) =
l
k
pij
∑ ∑ pij log pi p j
i =0 j =0
and again, analogously: MI (P , Q) =
C ∩ C  C ∩ C  log n n C C  C ∈P C ∈Q
∑ ∑
is the mutual information of two partitions [68]. Meilˇa [69] introduced the Variation of Information as V I = H (P ) + H (Q) − 2MI. Table A4. Information theorybased measures used in Table 3. All measures are based on Shannon’s deﬁnition of entropy. Again, n = V . Abbr.
Measure
MI
e.g., Vinh et al. [68]
NMI ϕ
Danon et al. [70]
Formula C ∩C  ∑C∈P ∑C ∈Q n MI , ϕ( H (P ),H (Q))
C ∩C  log n CC 
ϕ ∈ {min, max}
NMIΣ
Danon et al. [70]
2·MI H (P )+ H (Q)
VI
Meilˇa [69]
H (P ) + H (Q) − 2MI
P=P max 1.0 1.0 0.0
Appendix B.4. Summary As one can see, all three classes of measures rely mainly on set matching between node sets (clusters), as an alternative deﬁnition of N11 = ∑C∈P ∑C ∈Q (C∩2C ) shows [42]. The adjacency information of the graph is completely ignored. References 1. 2.
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SS symmetry Article
A MultiGranularity 2Tuple QFD Method and Application to Emergency Routes Evaluation Yanlan Mei, Yingying Liang * and Yan Tu * School of Management, Wuhan University of Technology, Wuhan 430070, China; [email protected] * Correspondence: [email protected] (Y.L.); [email protected] (Y.T.) Received: 23 September 2018; Accepted: 8 October 2018; Published: 11 October 2018
Abstract: Quality function deployment (QFD) is an effective approach to satisfy the customer requirements (CRs). Furthermore, accurately prioritizing the engineering characteristics (ECs) as the core of QFD is considered as a group decision making (GDM) problem. In order to availably deal with various preferences and the vague information of different experts on a QFD team, multigranularity 2tuple linguistic representation is applied to elucidate the relationship and correlation between CRs and ECs without loss of information. In addition, the importance of CRs is determined using the best worst method (BWM), which is more applicable and has good consistency. Furthermore, we propose considering the relationship matrix and correlation matrix method to prioritize ECs. Finally, an example about evaluating emergency routes of metro station is proposed to illustrate the validity of the proposed methodology. Keywords: quality function deployment; engineering characteristics; group decision making; 2tuple; metro station; emergency routes
1. Introduction In order to cope with intense global competitions, enterprises must design the highest quality products that satisfy the voice of customers (VOCs). Quality function deployment (QFD) is an effective method to map customer requirements (CRs) into engineering characteristics (ECs) in the area of product development [1] and construction industry [2]. The core of QFD is requirements conversion, moreover, the ﬁrst phase in house of quality (HOQ) mapping CRs to ECs becomes an essential procedure of implementing QFD [3]. Aiming at implementing QFD successfully, plenty of CRs should be acquired, and group decision making (GDM) should be adopted [4]. QFD consists of two major steps: collecting the CRs and mapping it to ECs, both of which are performed [5,6]. This paper focuses on how the ECs in QFD can be prioritized. There are plenty of methods to prioritize the ECs. Fuzzy set theory was widely employed to calculate the rankings of ECs under the circumstance of vagueness and impreciseness. Fuzzy multiple objective programming [7], fuzzy goal programming [8], fuzzy relationship and correlations [9], and expected valuebased method [10] are proposed to prioritize ECs. In addition, Geng et al. [11] integrated the analytic network process to QFD to reﬂect the initial importance weights of ECs. However, the problem is that they paid little attention to the GDM method, which can aggregate different experts’ preferences. For the purpose of reaching collective decisions, we combine GDM with QFD. Kwong et al. [6] put forward the fuzzy GDM method integrated with a fuzzy weighted average to rank ECs. Wang [12] adopted the method of aggregating technical importance rather than CRs to prioritize ECs. With respect to consensus, modiﬁed fuzzy clustering was presented so as to reach the consensus of the QFD team [13]. A twostage GDM was proposed to simultaneously solve the two types of uncertainties (i.e., human assessment on qualitative attributes as well as input
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information) underlying QFD [4]. However, due to varying personal experience and knowledge, the input information of experts presented with multiformat or multigranular linguistic preferences makes prioritizing ECs more difﬁcult. Therefore, some scholars have focused on the GDM approach based on multigranularity linguistic environments [14–17]. Xu [18,19] analyzed multiple formats’ preferences and provided an approach integrating information in the context of GDM. It is noteworthy that multigranularity evaluation should be analyzed. The correlation between CRs and ECs inﬂuencing on the relationship becomes ignored and simpliﬁed in the current study. In addition, the linguistic accuracy remains to be discussed. Considering that the 2tuple linguistic representation can increase the information of precision [20,21]. In order to ﬁll the gap, it is necessary that the QFD methodology is extended with a 2tuple linguistic environment so as to lessen the loss of information and obtain accurate value of ECs. In addition, decision makers may have different knowledge and experience in the process of group decision making, and they may then adopt different linguistic labels to describe the same decisionmaking problems. This process is denoted as multigranular linguistic information, which conforms to the actual decisionmaking process. Therefore, we allow decision makers to employ multigranular linguistic information, i.e., the linguistic term set has different granularities. A majority of methods deal with multigranular linguistic information. Herrera et al. proposed the deﬁnition of a basic linguistic term set, and then different linguistic labels can be uniﬁed based on a basic linguistic term set [22]. In addition, some transformation methods based on the linguistic hierarchy and extended linguistic hierarchy were presented and applied to a plenty of decisionmaking problems [23,24]. Among these approaches, the method considering linguistic hierarchy is more ﬂexible and convenient to carry out. In this paper, we adopt this method to deal with the problem of multigranular linguistic evaluation. For determining the weight of CRs, we adopt the best–worst method (BWM) in this paper. This method has good consistency and is easier to implement [25,26]. Our contributions lie in using the BWM to determine the importance of CRs and integrate the correlations matrix with the relationship matrix based on a compromise idea, where experts can express their thoughts in different granularities. In this paper, a GDM approach is integrated with QFD to solve different preferences and prioritize ECs. The multigranularity 2tuple linguistic information to reﬂect the attitudes of different experts is employed. This paper is organized as follows: In Section 2, a 2tuple multigranularity linguistic representation model, linguistic hierarchies, and a 2tuple linguistic weighted geometric Bonferroni mean (2TLWGBM) operator are presented. In Section 3, the BWM is applied to compute the weight of CRs, and a novel GDM approach to prioritize ECs is proposed. An illustrated example about metro stations is provided in Section 4 to demonstrate the applicability of this method. Ultimately, conclusions and future research are marked in Section 5. 2. Preliminaries In this section, we introduce some basic knowledge about QFD, 2tuple representation and the 2TLWGBM Operator. 2.1. The Basic Knowledge on QFD A fourphase QFD model is employed to translate the VOCs to ECs, which consists of Product Planning, Part Deployment, Process Planning, and Process and Quality control [3]. The ﬁrst phase is to collect customer requirements for the product called WHATs and then to transform these needs into ECs called HOWs. This phase is so fundamental in product development that the corresponding QFD transformation matrix referred to the HOQ (Figure 1). The HOQ links customer needs to the development team’s technical responses, so we focus on this phase in order to translate different preference of customers and experts to prioritize ECs. In this paper, we ﬁrst take the relationship between CRs and ECs into consideration. In order to transform the importance of CRs into ECs, the correlation of CRs and ECs is introduced to modify the initial relationship afterward.
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Figure 1. House of Quality (HOQ).
2.2. The 2Tuple Linguistic Representation There are numerous formats for representing preference such as linguistic, numerical and 2tuple linguistic representation. Compared with other forms, 2tuple linguistic representation makes the assessment more precise and without a loss of information [20]. Next, we will introduce some basic knowledge about 2tuple representation. 3 4 Deﬁnition 1 [27]. Assuming S = s1 , s2 , · · · , s g is a linguistic term set and β ∈ [0, g] represents the consequence of a symbolic aggregation operation. Afterwards, the 2tuple is expressed as the equivalence to β as follows: (1) Δ : [0, g] → S × [−0.5, 0.5) ( si , i = round( β) Δ( β) = (si , α), with (2) α = β − i, α = [−0.5, 0.5) where round (·) represents the usual round function, si has the closest index label to β, and α is the value of the symbolic translation. 3 4 Deﬁnition 2 [27]. Let S = s1 , s2 , · · · , s g be a linguistic term set and (si , αi ) be a 2tuple. There is always a − 1 function Δ that can be deﬁned, such that, from a 2tuple (si , αi ), its equivalent numerical value β ∈ [0, g] ⊂ R can be obtained, which is described as follows: Δ−1 :→ S × [−0.5, 0.5) → [0, g]
(3)
Δ −1 ( s i , α i ) = i + α i = β
(4)
Deﬁnition 3 [28]. There are 2tuples x = {(s1 , α1 ), (s2 , α2 ), · · · , (sn , αn )} . Their arithmetic mean is expressed as: 1 n −1 (s, α) = Δ Δ (ri , αi ) , s ∈ S, α ∈ [−0.5, 0.5) (5) ∑ n i =1 292
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Deﬁnition 4 [28]. Let (si , αi ) and (s j , α j ) be two 2tuple linguistic variables. Their granularities are both g, and the distance between them is described as follows: & −1 & & Δ ( i + α i ) − Δ −1 ( j + α j ) & d((si , αi ), (s j , α j )) = g
(6)
Deﬁnition 5 [28]. Let x = {(s1 , α1 ), (s2 , α2 ), · · · , (sn , αn )} be a set of 2tuples and (s, α) be the arithmetic mean of these 2tuples. The degree of similarity is expressed as sim((sπ ( j) , απ ( j) ), (s, α)) = 1 −
d((sπ ( j) , απ ( j) ), (s, α)) n
∑ d((sπ ( j) , απ ( j) ), (s, α))
, j = 1, 2, · · · , n
(7)
j =1
Deﬁnition 6 [23]. Let LH = ∪l (t, n(t)), which is the union of all level t, a linguistic hierarchy whose t
#
$ n(t) n(t) n(t) linguistic term set is = s0 , s1 , · · · , sn(t)−1 . Furthermore, different granularities reﬂect different preferences under the circumstance of evaluating. The transformation function (TF) between level t and level t is deﬁned as Sn(t)
TFtt : l (t, n(t)) → l (t, n(t)) n(t) Δ−1 (si , αn(t) ) · (n(t) − 1) n(t) TFtt (si , αn(t) ) = Δ n(t) − 1
(8)
where t and t represent different levels of linguistic hierarchy. Note 1. The TF can implement the transformation between different granularities and further achieve a uniﬁed linguistic label. Without loss of generality, the transformation usually is carried out from the lower granularity to higher granularity in the process of transformation, i.e., the level t usually corresponds to the maximum granularity. 2.3. The 2TLWGBM Operator There are numerous operators to aggregate information in different linguistic environments, such as hesitant fuzzy Maclaurin symmetric mean Operators [29], 2tuple linguistic Muirhead mean operators [30], 2tuple linguistic Neutrosophic number Bonferroni mean operators [31], and hesitant 2tuple linguistic prioritized weighted averaging aggregation operator [32] in the context of the 2tuple environment. In view of the Bonferroni mean (BM) operator capturing the interrelationship between input information and ranking ECs under a 2tuple environment, so the 2TLWGBM operator [33] will be applied to prioritize the sequence of ECs. BM is deﬁned as follows: Deﬁnition 7 [33]. Let p, q ≥ 0 and ai (i = 1, 2, · · · , n) be a series of nonnegative numbers. Then the BM operator is deﬁned as ⎛
BM
p,q
⎞
⎜ ⎟ ⎜ ⎟ n ⎜ ⎟ 1 p q ⎟ a ( a1 , a2 , · · · , a n ) = ⎜ a ∑ ⎜ n ( n − 1) i j⎟ ⎜ ⎟ i, j = 1 ⎝ ⎠ i = j
1 p+q
(9)
Deﬁnition 8 [33]. Let x = {(r1 , a1 ), (r2 , a2 ), · · · , (rn , an )} be a set of 2tuple and p, q ≥ 0. In addition, w = (w1 , w2 , · · · , wn ) T is the weight vector of x, where wi > 0 (i = 1, 2, · · · , n) represents the importance degree of (ri , ai ) (i = 1, 2, · · · , n), and ∑in=1 wi = 1. The 2TLWGBM operator is then expressed as
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p,q
2TLWGBMw ((r1 , a1 ), (r2 , a2 ), · · · , (rn , an )) ⎛ ⎞ 1 ⎞ ⎛ n ( n −1) ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ n ⎜ 1 ⎜ ⎟ −1 wi w j ⎟ −1 ⎟ p Δ (ri , ai ) + q Δ (r j , a j ) = Δ⎜ ⎟ ∏ ⎜ p+q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ i, j = 1 ⎝ ⎠ i = j
(10)
For the sake of simplicity, it can be set p = q = 1, the aggregation operator is indicated as 1,1 ((r1 , a1 ), (r2 , a2 ), · · · , (rn , an )) 2TLWGBMw ⎛ ⎛ ⎞ 1 ⎞ n ( n −1) ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ n ⎜1⎜ ⎟ wi −1 w j ⎟ ⎟ − 1 ⎟ ( r , a ) + ( r , a ) Δ = Δ⎜ Δ ⎟ ⎜ ∏ i i j j ⎜2⎜ ⎟ ⎟ ⎜ ⎝ i, j = 1 ⎟ ⎠ ⎝ ⎠ i = j
(11)
Note 2. Although a majority of aggregation operators have been proposed in recent years, the 2TLWGBM operator has some merits in prioritizing ECs. On the one hand, this operator considers the relevance, which accords with the relationship and correlation between CRs and ECs. On the other hand, it is more ﬂexible owing to the parameter p and q, which makes it more suitable for different decision makers. 3. A Group DecisionMaking Approach to Prioritize ECs 3.1. Determine the Importance of CRs Based on BWM Best worst method (BWM) is a MCDM method possessing the advantages in aspects of reaching the consistency and simplifying the calculation with respect to AHP. The core idea of BWM is constructing comparisons relationships between the best attribute (and the worst attribute) to the other attributes. Additionally, an optimization model established ground on consistency is solved to obtain the optical weights. Owing to simple operation and calculation, the BWM is synthesized to determine the importance of CRs. The steps are listed as follows: Step 1. CRs {CR1 , CR2 , · · · , CRn } are chosen, as are the best and the worst CR. The best CR is then compared with the other CRs using Number 1–9 is constructed. The besttoothers (BO) vector AB = (αB1 , αB2 , · · · , αBn ) is represented where αBj describes the preference of the best CR over CRj . Similarly, the Otherstoworst (OW) vector AW = (α1W , α2W , · · · , αnW )T is represented where αjW describes the preference of CRj over the worst CR. Step 2. The optimal weights of CRs are obtained. The optimization model is established to minimize the maximum the difference {wB − αBj wj } and {wj − αjW ω W }.
^
PLQ PD[ M Z% D%M Z M Z M D M: Z: VW ¦ Z M
(Model 1)
M
Z M t M
`
Q
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Model 1 can be transformed into a linear programming model as follows:
PLQ [ VW Z% D%M Z M d [ M Z M D M: Z: d [ M
¦Z
M
Q Q (Model 2)
M
Z M t M
Q
Model 2 is solved to obtain the optimal importance of CRs (w1∗ , w2∗ , · · · , wn∗ ) and ξ * . Alternatively, the bigger ξ * demonstrates the higher consistency ratio provided by customers. The consistency ratio can be calculated by the proportion between ξ * and max ξ (Consistency Index). Consistency Ratio =
ξ∗ ξ∗ = max ξ Consistency Index
(12)
where the max ξ is determined according to (αBW − ξ) × (αBW − ξ) = (αBW + ξ) and αBW ∈ {1, 2, · · · , 9}. The consistency index is listed in Table 1. Table 1. Consistency index. αBW
1
2
3
4
5
6
7
8
9
Consistency index
0.00
0.44
1.00
1.63
2.30
3.00
3.73
4.47
5.23
3.2. A Group DecisionMaking Approach to Prioritize ECs In this section, the steps of GDM for multigranularity 2tuple linguistic preference to prioritize ECs in QFD are given as follows: Step 3. Different multigranularity linguistic preferences are obtained. Suppose the experts EPk (k = 1, 2, · · · , t) in QFD product research or design team give the relationship between CRi (i = 1, 2, · · · , n) and ECj ( j = 1, 2, · · · , s) based on different multigranularities. The kth expert’s linguistic term set $ # and evaluation matrix respectively denoted as Sn(t)k =
n(t)k
and Rk = rij )n×s rij ∈ Sn(t)k , which is transformed into 5 k = r n ( t ) k , 0) n × s , r n ( t ) k ∈ S n ( t ) k . 2tuple linguistic evaluation matrix R ij ij si
i = 0, 1, · · · , n(t) − 1
Step 4. Different multigranularity linguistic preferences are uniﬁed. $ # n(t) To begin with, a basic linguistic term set Sn(t)u = si u i = 0, 1, · · · , n(t) − 1 can be chosen, and the relationship matrix can then be transformed applying Equation (8) so as to make 2tuple linguistic representation reach the same granularity. For instance, the kth expert’s judgement matrix is 5 k = r n(t)u , α n(t)u )n×s , r n(t)u ∈ S n(t)u . transformed as R ij
ij
ij
Step 5. All the evaluation matrices are aggregated. All the evaluation matrices uniformed are aggregated with 2TLWGBM operator in virtue of Equation (10) into Rij (i = 1, 2, · · · , n ; j = 1, 2, · · · , s). Furthermore, the new matrix represents ultimate relationship matrix between CRs and ECs in essence. Step 6. The relationship between CRs and ECs is modiﬁed based on a compromise idea. After establishing the aggregation matrix, experts give the correlations among CRs and ECs and the initial HOQ can be obtained, which reﬂects the relationship Rij between CRi and 295
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ECj ,i = 1, 2, · · · , n ; j = 1, 2, · · · , s.It is indispensable that the QFD team estimates the correlations between CRs (i.e., Liξ (ξ = i, ξ = 1, 2, · · · , n)) and ECs (i.e., Tiθ (θ = i, θ = 1, 2, · · · , s)) using 2tuple n(t)
based on the basic linguistic set Si u . Considering that the assessment result of Liξ (ξ = i, ξ = 1, 2, · · · , n) and Tiθ (θ = i, θ = 1, 2, · · · , s) has an effect on the initial aggregation matrix of relationship Rij with respect to CRs and ECs, a higher Liξ (ξ = i, ξ = 1, 2, · · · , n) or Tiθ (θ = i, θ = 1, 2, · · · , s) implies a beneﬁt to Rij . Consequently, the correlations are taken into account when modifying the relationships between CRs and ECs. In the process of adjustment, Equation (13) is applied to integrate Liξ (ξ = i, ξ = 1, 2, · · · , n) and Tiθ (θ = i, θ = 1, 2, · · · , s) into Rij . Furthermore, the modiﬁed relationship is computed using the formula as follows: Rij
=Δ
V
∏Δ
−1
v =1
(sm , αm )
γv
, i = 1, 2, · · · , n ; j = 1, 2, · · · , s
(13)
# n(t) n(t) n(t) n(t) where Δ−1 (rm , am ) is stemming from the set S = Δ−1 Rij (rij u , αij u ), Δ−1 Liξ riξ u , αiξ u ), $ n(t) n(t) Δ−1 Tjθ (r jθ u , α jθ u ) ξ = i, θ = j. In addition, the weight of γv is corresponding to the proportion
of Δ−1 (rm , am ). For the sake of reducing the impact from subjectivity, unduly high or unduly low preference values in the correlation matrices are supposed to possess a low weight under the circumstances. That means only moderated assessment giving a higher weight has a small deviation from the true value, which might be advocated in the process of evaluation. Therefore, the weight can be determined by Equation (14). sim((sm , αm ), (s, α))
γv =
V
(14)
∑ sim((sm , αm ), (s, α))
v =1
Step 7. Integrated ECs priorities are determined. On account of the inconformity of representation, the 2tuple linguistic form of the relationship matrix, and the numerical value of CRs importance, the integrated ECs priority STCj ( j = 1, 2, · · · , s) is calculated by Equation (11). Step 8. Basic priority of ECs is conﬁrmed. n(t)
n(t)
The linguistic distance d((sTCj , αn(t) )(smin , αn(t) )) can be adopted to measure the importance n(t)
degree, where (smin , αn(t) ) is the minimum value of linguistic term set. Furthermore, the measurement of 2tuple linguistic distance decides the importance of ECj ( j = 1, 2, · · · , s). Therefore, the normative value of basic priority bpr j is computed as follows: n(t)
bpr j =
n(t)
d((sECj , αn(t) )(smin , αn(t) )) s
n(t)
n(t)
∑ d((sECj , αn(t) )(smin , αn(t) ))
j =1
Step 9. End. The ﬂow chart of the whole procedures is shown in Figure 2.
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(13)
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Figure 2. Group decision making for multigranularity 2tuple linguistic preference to prioritize engineering characteristics in quality function deployment.
4. Case Study 4.1. Background The Wuhan metro station is the most common twoﬂoor island structure, which consists mainly of a platform and a station hall. The underground ﬂoor is the station hall ﬂoor. As shown in Figure 3, the metro station has four main exits and one reserved outlet for docking with the shopping mall and ﬁre curtains are installed at each exit. Therefore, when a crowd passes through the ﬁre curtain in the emergency evacuation process, they have reached the safe area. The station hall ﬂoor has four automatic ticket checkers and two emergency dedicated channels. In emergency situations, an automatic ticket checking machine and emergency dedicated channels are in open state. The second underground ﬂoor is the platform layer. When an emergency occurs on the platform layer, the crowd must ﬁrst ascend to the station hall layer and then evacuate through the safety exit.
Figure 3. The structure of metro station in Wuhan.
Taking regional S as an example, we analyze the inﬂuence factors that have an effect on the evacuation route planning in this area. Five CRs and ECs are selected in order to determine the weight degree of ECs, which can be a basic of evaluating emergency routes. 297
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CR1 CR2 CR3 CR4 CR5
: : : : :
Expected evacuation time Crowd density Risk level in the region Possibility of congestion Evacuation capability
: : : : :
EC1 EC2 EC3 EC4 EC5
Number of evacuees per unit time Managerial capability Risk level of disaster Organizational situation Evacuation equipment
4.2. Implementation Step 1. The evaluation CRs relationships by passengers are shown in Tables 2 and 3. Table 2. Besttoothers (BO) vector for passengers. Passengers 1 2 3 4 5
Best
CR1
CR2
CR3
CR4
CR5
CR2 CR2 CR1 CR3 CR1
3 5 1 4 1
1 1 2 3 3
5 4 7 1 9
9 9 5 7 6
7 8 9 9 8
Table 3. Otherstoworst (OW) vector for passengers. Passengers
1
2
3
4
5
Worst CR1 CR2 CR3 CR4 CR5
CR4 6 9 5 1 4
CR4 4 9 7 1 3
CR5 9 8 2 5 1
CR5 4 7 9 2 1
CR3 9 7 1 6 3
Step 2. The importance of CRs is respectively computed as 0.302, 0.359, 0.187, 0.082 and 0.070, which is determined by the average value by passengers. For example, the model by ﬁrst passenger is established as follows: min ξ s.t. w2 − 3w1  ≤ ξ, w2 − 5w3  ≤ ξ, w2 − 9w4  ≤ ξ, w2 − 7w5  ≤ ξ, w1 − 6w4  ≤ ξ, w3 − 5w4  ≤ ξ, w5 − 4w4  ≤ ξ, ∑ w j = 1, 5
w j ≥ 0, j = 1, 2, · · · , 5. The parameter ξ is obtained as 0.12, and the consistency ratio can be then computed using Equation (12) as 0.023, which indicates it has good consistency. Step 3. In order to determine the basic priority of these ECs, three experts EP1 , EP2 , EP3 evaluate the importance of ECs according to CRs given as below (Tables 4–6). They represent preference 3 4 3 4 9 7 by using the different linguistic term sets Si 1 = s70 , s71 , s72 , s73 , s74 , s75 , s76 Si52 = s50 , s51 , s52 , s53 , s54 Si 3 = 3 9 9 9 9 9 9 9 9 94 s0 , s1 , s2 , s3 , s4 , s5 , s6 , s7 , s8 .
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Table 4. Evaluation matrix R1 for EP1 .
CR1 CR2 CR3 CR4 CR5
EC1
EC2
EC3
EC4
EC5
s751 s711 s721 s731 s711
s761 s751 s741 s751 s741
s701 s711 s741 s711 s711
s741 s731 s711 s761 s741
s701 s741 s711 s751 s751
Table 5. Evaluation matrix R2 for EP2 .
CR1 CR2 CR3 CR4 CR5
EC1
EC2
EC3
EC4
EC5
s532 s512 s512 s522 s502
s542 s532 s522 s532 s532
s502 s522 s532 s512 s512
s532 s522 s512 s542 s522
s512 s532 s512 s532 s532
Table 6. Evaluation matrix R3 for EP3 .
CR1 CR2 CR3 CR4 CR5
EC1
EC2
EC3
EC4
EC5
s963 s923 s903 s933 s913
s983 s973 s963 s973 s973
s913 s923 s963 s923 s913
s953 s963 s923 s983 s973
s903 s963 s943 s963 s973
Step 4. Three evaluation matrices are transformed into 2tuple representation in Tables 7–9. 51 for EP1 . Table 7. 2tuple linguistic evaluation matrix R CR1 CR2 CR3 CR4 CR5
EC1
s 71 , 0 5 s 71 , 0 1 s 71 , 0 2 s71 , 0 3 s711 , 0
EC2
s 71 , 0 6 s 71 , 0 5 s 71 , 0 4 s 71 , 0 5 s741 , 0
EC3
s 71 , 0 0 s 71 , 0 1 s 71 , 0 4 s 71 , 0 1 s711 , 0
EC4
s 71 , 0 4 s 71 , 0 3 s 71 , 0 1 s 71 , 0 6 s741 , 0
EC5 s701 , 0 s741 , 0 s711 , 0 s751 , 0 s751 , 0
52 for EP2 . Table 8. 2tuple linguistic evaluation matrix R CR1 CR2 CR3 CR4 CR5
EC1
s 52 , 0 3 s 52 , 0 1 s 52 , 0 1 s52 , 0 2 s502 , 0
EC2
s 52 , 0 4 s 52 , 0 3 s 52 , 0 2 s 52 , 0 3 s532 , 0
EC3
s 52 , 0 0 s 52 , 0 2 s 52 , 0 3 s 52 , 0 1 s512 , 0
299
EC4
s 52 , 0 3 s 52 , 0 2 s 52 , 0 1 s 52 , 0 4 s522 , 0
EC5 s512 , 0 s532 , 0 s512 , 0 s532 , 0 s532 , 0
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53 for EP3 . Table 9. 2tuple linguistic evaluation matrix R CR1
CR2
EC1 s963 , 0 s923 , 0
CR3
s903 , 0
CR4
s933
CR5
s913 , 0
EC2 s983 , 0 s973 , 0 s963 , 0 s973 , 0 s973 , 0
EC3 s913 , 0 s923 , 0 s963 , 0 s923 , 0 s913 , 0
EC4 s953 , 0 s963 , 0 s923 , 0 s983 , 0 s973 , 0
EC5 s903 , 0 s963 , 0 s943 , 0 s963 , 0 s973 , 0
Step 5. The aggregation of all the evaluation matrices in Tables 9–11 applying 2TLWGBM operator in Equation (11) into Rij is shown in Table 12. 51 for EP1 . Table 10. The transformed 2tuple linguistic evaluation matrix R
CR1 CR2 CR3 CR4 CR5
EC1
EC2
EC3
EC4
EC5
(s971 , −0.33) (s911 , 0.33) (s931 , −0.33)
(s981 , 0) 91 (s7 , −0.33) (s951 , 0.33)
(s901 , 0) (s911 , 0.33) (s951 , 0.33)
(s951 , 0.33) (s941 , 0) (s911 , 0.33 ) s983 , 0
(s901 , 0) (s951 , 0.33) (s911 , 0.33)
(s911 , 0.33)
(s951 , 0.33)
(s911 , 0.33)
(s951 , 0.33)
(s941 , 0)
(s971 , −0.33)
(s911 , 0.33)
(s971 , −0.33) (s971 , −0.33)
52 for EP2 . Table 11. The transformed 2tuple linguistic evaluation matrix R
CR1 CR2 CR3 CR4 CR5
EC1
EC2
EC3
EC4
EC5
(s962 , 0) (s922 , 0) (s922 , 0) (s942 , 0) (s902 , 0)
(s982 , 0) (s962 , 0) (s942 , 0) (s962 , 0) (s962 , 0)
(s901 , 0) (s942 , 0) (s962 , 0) (s922 , 0) (s922 , 0)
(s962 , 0) (s941 , 0) (s922 , 0) (s982 , 0) (s942 , 0)
(s922 , 0) (s962 , 0) (s922 , 0) (s962 , 0) (s962 , 0)
Table 12. The aggregation of all the evaluation matrices.
CR1 CR2 CR3 CR4 CR5
EC1
EC2
EC3
EC4
EC5
(s92 , −0.14) (s91 , 0.23) (s91 , −0.23) (s92 , −0.48) (s91 , −0.35)
(s92 , 0.05) (s92 , −0.08) (s92 , −0.23) (s92 , −0.08) (s92 , −0.1)
(s90 , 0) (s91 , 0.33) (s92 , −0.16) (s91 , 0.23) (s91 , 0.1)
(s92 , −0.23) (s92 , −0.26) (s91 , 0.23) (s92 , 0.05) (s92 , −0.17)
(s90 , 0) (s92 , −0.16) (s91 , 0.41) (s92 , −0.14) (s92 , −0.08)
Step 6. On the basic of different knowledge and experience, three experts adopt their own linguistic representations to evaluate correlations between CRs and ECs. These matrices are then aggregated in the same way as the fourth step. Consequently, the initial HOQ is shown in Figure 4.
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Figure 4. The 2tuple initial HOQ.
In Figure 4, the correlations between CRs and ECs are computed in the same way as the relationships between CRs and ECs are treated. Apparently, an appropriate relationship matrix should take correlations into account, so the modiﬁed relationship in virtue of Equations (5)–(7), (12), and (13) is obtained. The result is illustrated in Figure 5. We take the relationship between CR1 and EC1 for an example, the process of calculation is demonstrated as follow
(s, α) = '( 13 ('−1 (s92 , −0.14) + ('−1 (s92 , −0.32) + ('−1 (s91 , 0.39)) = (s92 , −0.36) & −1 & &' (2 − 0.14) − '−1 (2 − 0.36)& d((s92 , −0.14), (s92 , −0.36)) = = 0.024 9 Similarly, d((s92 , −0.32), (s92 , −0.36)) = 0.004, d((s91 , 0.39), (s92 , −0.36)) = 0.028 sim((s92 , −0.14), (s92 , −0.36)) = 1 −
0.024 = 0.571 0.024 + 0.004 + 0.028
Similarly, sim((s92 , −0.32), (s92 , −0.36)) = 0.928, sim((s91 , 0.39), (s92 , −0.36)) = 0.5 We then compute the weight γv by Equation (13) γ(s92 , −0.14) =
0.571 = 0.286 0.571 + 0.928 + 0.5
In the same way, γ(s92 , −0.32) = 0.464, γ(s91 , 0.39) = 0.25 The modiﬁed relationship is expressed = '('−1 (s92 , −0.14) R11
0.286
∗ '−1 (s92 , −0.32)
301
0.464
∗ '−1 (s91 , 0.39)
0.25
) = (s92 , −0.35)
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Figure 5. The 2tuple modiﬁed HOQ.
Step 7. After obtaining the modiﬁed matrix, the importance of CRs should be integrated to reach the ﬁnal relationships between CRs and ECs. The result is presented in Table 13. Therefore, the rank of integrated ECs priority is EC2 ( EC4 ( EC1 ( EC5 ( EC3 . Table 13. The integrated ECs priority.
Priority
EC1
EC2
EC3
EC4
EC5
(s91 , 0.07)
(s91 , 0.13)
(s91 , −0.18)
(s91 , 0.11)
(s91 , −0.16)
Step 8. The basic priority of ECs is computed according to Equations (7) and (14) and Table 13. n(t)
The minimum value of linguistic term set is (smin , αn(t) ) = (s90 , 0). The ultimate weights of ECs are (s90 , 0.215) (s90 , 0.228) (s90 , 0.165) (s90 , 0.223) (s90 , 0.169). Step 9. End. 4.3. Managerial Tips The outcomes of this study are beneﬁcial to planning and selecting the appropriate emergency routes. Moreover, the ranking result can be outlined that decision makers should be paid more attention to management ability. The result indicates that crowd density has a signiﬁcant inﬂuence on emergency route evaluation. Subsequently decision makers should concentrate on these two aspects in order to design and select emergency routes. In addition, the proposed model is sufﬁcient robust and could be easily implemented in practices for GDM problems. DMs can choose their linguistic preference to evaluate the correlation and relationship between CRs and ECs. Furthermore, the importance of ECs can be adjusted appropriately according to the actual circumstance.
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5. Conclusions and Future Research A systematic GDM approach for prioritizing ECs in QFD under the multigranularity 2tuple linguistic environment is proposed in this paper. The provided method allows experts from QFD team to evaluate the relationship and correlations between CRs and ECs in accordance with their experience and preference. For the sake of guaranteeing accurate information, the 2tuple linguistic representation addressing the vague and imprecise information is utilized. Based on the linguistic hierarchy, different granularities originating from different experts are translated into a basic linguistic term we set in advance. The BWM is applied to determine the importance of CRs, which is simple and quick to represent customers’ advice. BM can capture interrelationships among the aggregated information by taking the conjunction among each pairs of aggregated arguments, for instance, correlations among CRs. Therefore, the 2TLWGBM operator is applied to aggregate the evaluation matrix and the importance of CRs. In addition, correlations could have an impact on relationship between CRs and ECs. A modiﬁed matrix reﬂecting the inﬂuence is determined in this paper. Compared with other approaches in terms of calculating weight, a method that can lessen the subjectivity of assessment is put forward. Finally, a case study has been calculated and is presented to verify the effectiveness of the proposed method. In this study, prioritizing ECs in QFD is extended to 2tuple linguistic environment, in which all evaluation matrices from experts are represented by 2tuple. For one thing, an appropriate and applicable BM operator is employed to deal with the aggregation problem, which should be suitable for accurately prioritizing ECs in QFD. Moreover, the degree of similarity is introduced to determine the weight that responds to the effect of correlations, which could obtain a more objective modiﬁed matrix. In future research, the proposed method can be applied to supplier selection, green buildings and new product development. In addition, other GDM approaches can be integrated into QFD to rank the ECs, and consensus can be considered. A more reasonable aggregation operator should be developed and applied to QFD. In real life, plenty of problems might be complex and changeful. Establishing a dynamic HOQ is necessary. Author Contributions: Y.M. drafted the initial manuscript and conceived the model framework. Y.L. provided the relevant literature review and the illustrated example. Y.T. revised the manuscript and analyzed the data. Funding: This research is supported by the National Social Science Foundation of China (Project No. 15AGL021), the National Natural Science Foundation of China (NSFC) under Project 71801177, and the Project of Humanities and Social Sciences (18YJC63016). Acknowledgments: The authors would like to thank the anonymous referees and academic editor for their very valuable comments. Conﬂicts of Interest: The author declares no conﬂict of interest.
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The Structure Theorems of PseudoBCI Algebras in Which Every Element is QuasiMaximal Xiaoying Wu 1 and Xiaohong Zhang 1,2, * 1 2
*
Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China; [email protected] Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China Correspondence: [email protected] or [email protected]
Received: 19 September 2018; Accepted: 29 September 2018; Published: 8 October 2018
Abstract: For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. PseudoBCI algebra is a class of nonclassical logic algebras, which is closely related to various noncommutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudoBCI algebras in which every element is quasimaximal (call it QMpseudoBCI algebras in this paper). First, the new notions of quasimaximal element and quasileft unit element in pseudoBCK algebras and pseudoBCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QMpseudoBCI algebra is proved: every QMpseudoBCI algebra is a KGunion of a quasialternating BCKalgebra and an antigroup pseudoBCI algebra. Third, the new notion of weak associative pseudoBCI algebra (WApseudoBCI algebra) is introduced and the following result is proved: every WApseudoBCI algebra is a KGunion of a quasialternating BCKalgebra and an Abel group. Keywords: fuzzy logic; pseudoBCI algebra; quasimaximal element; KGunion; quasialternating BCKalgebra
1. Introduction In the study of tnorm based fuzzy logic systems [1–9], algebraic systems (such as residuated lattices, BLalgebras, MTLalgebras, pseudoBL algebras, pseudoMTL algebras, et al.) play an important role. In this paper, we discuss pseudoBCI/BCK algebras which are connected with noncommutative fuzzy logic systems (such that noncommutative residuared lattices, pseudoBL/pseudoMTL algebras). BCKalgebras and BCIalgebras were introduced by Is´eki [10] as algebras induced by Meredith’s implicational logics BCK and BCI. The name of BCKalgebra and BCIalgebra originates from the combinatories B, C, K, I in combinatory logic. The notion of pseudoBCK algebra was introduced by G. Georgescu and A. Iorgulescu in [11] as a noncommutative extension of BCKalgebras. Then, as common generalization of pseudoBCK algebras and BCIalgebras, W.A. Dudek and Y.B. Jun introduced the concept of pseudoBCI algebra in [12]. In fact, there are many other nonclassical logic algebraic systems related to BCK and BCIalgebras, such as BCCalgebra, BZalgebra and so forth, some monographs and papers on these topics can be found in [7–9,13–18]. PseudoBCIalgebras are algebraic models of some extension of a noncommutative version of the BCIlogic, the corresponding logic is called pseudoBCI logic [19]. P. Emanovský and J. Kühr studied some properties of pseudoBCI algebras, X.L. Xin et al. [20] investigated monadic pseudo BCIalgebras and corresponding logics and some authors discussed the ﬁlter (ideal) theory of pseudoBCI algebras [21–28]. Moreover, some notions of period, state and soft set are applied to pseudoBCI algebras [29–31]. In this paper, we further study the structure characterizations of pseudoBCI algebras. By using the notions of quasimaximal element, quasileft unit element, KGunion and direct product, we give Symmetry 2018, 10, 465; doi:10.3390/sym10100465
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the structure theorem of the class of pseudoBCI algebras in which every element is quasimaximal (call they QMpseudoBCI algebras). Moreover, we introduce weak associative property in pseudoBCI algebras, discuss basic properties of weak associative pseudoBCI algebra (WApseudoBCI algebra) and establish the structure theorem of WApseudoBCI algebra. It should be noted that the original deﬁnition of pseudoBCI/BCK algebra is different from the deﬁnition used in this paper. They are dual. We think that the logical semantics of this algebraic structure can be better represented by using the present deﬁnition. 2. Preliminaries Deﬁnition 1 ([10,16]). An algebra (A; →, 1) of type (2,0) is called a BCIalgebra if the following conditions are satisﬁed for all x, y, z from A: (1) (2) (3) (4) (5)
x → y ≤ ( y → z ) → ( x → z ), x ≤ ( x → y) → y, x ≤ x, x ≤ y, y ≤ x imply x = y, where x ≤ y means x → y = 1 . An algebra (A; →, 1) of type (2,0) is called a BCKalgebra if it is a BCIalgebra and satisﬁes: x → 1 = 1, ∀x ∈ A.
Deﬁnition 2 ([10,16]). A BCKalgebra (A; →, 1) is called bounded if there exists unique element 0 such that 0 → x = 1 for any x ∈ A. Deﬁnition 3 ([13,14]). A BCKalgebra (A; →, 1) is called quasialternating BCKalgebra if it satisﬁes the following axiom: ∀ x, y ∈ X, x = y implies x → y = y. Deﬁnition 4 ([9,11]). A pseudoBCK algebra is a structure (A; ≤, →, , 1), where “≤” is a binary relation on A, “→” and “” are binary operations on A and “1” is an element of A, verifying the axioms: for all x, y, z ∈ A, (1) (2) (3) (4) (5) (6)
x → y ≤ ( y → z ) ( x → z ), x y ≤ ( y z ) → ( x z ), x ≤ ( x → y) y, x ≤ ( x y) → y x ≤ x, x ≤ 1, x ≤ y, y ≤ x ⇒ x = y, x ≤ y x → y = 1 ⇔ x y = 1.
If (A; ≤, →, , 1) is a pseudoBCK algebra satisfying x → y = x y for all x, y ∈ A, then (A; →, 1) is a BCKalgebra. Proposition 1 ([9,11]). Let (A; ≤, →, , 1) be a pseudoBCK algebra, then A satisfy the following properties (∀ x, y, z ∈ A): (1) (2) (3) (4) (5) (6) (7) (8) (9)
x ≤ y ⇒ y → z ≤ x → z, y z ≤ x z x ≤ y, y ≤ z ⇒ x ≤ z, x ( y → z ) = y → ( x z ), x ≤ y → z ⇔ y ≤ x z, x → y ≤ ( z → x ) → ( z → y ), x y ≤ ( z x ) ( z y ), x ≤ y → x, x ≤ y x, 1 → x = x, 1 x = x, x ≤ y ⇒ z → x ≤ z → y, z x ≤ z y, ((y → x ) x ) → x = y → x, ((y x ) → x ) x = y x. 307
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Deﬁnition 5 ([[12]). A pseudoBCI algebra is a structure (A; ≤, →, , 1), where “≤” is a binary relation on A, “→” and “” are binary operations on A and “1” is an element of A, verifying the axioms: for all x, y, z ∈ A, (1) (2) (3) (4) (5)
x → y ≤ ( y → z ) ( x → z ), x y ≤ ( y z ) → ( x z ), x ≤ ( x → y) y, x ≤ ( x y) → y, x ≤ x, if x ≤ y and y ≤ x, then x = y, x ≤ y iff x → y = 1 iff x y = 1. Note that, every pseudoBCI algebra satisfying x → y = x y for all x, y ∈ A is a BCIalgebra.
Proposition 2 ([12,22,24]). Let (A; ≤, →, , 1) be a pseudoBCI algebra, then A satisfy the following properties (∀ x, y, z ∈ A): (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
if 1 ≤ x, then x = 1, if x ≤ y, then y → z ≤ x → z and y z ≤ x z, if x ≤ y and y ≤ z, then x ≤ z, x ( y → z ) = y → ( x z ), x ≤ y → z, iff y ≤ x z x → y ≤ ( z → x ) → ( z → y ), x y ≤ ( z x ) ( z y ), if x ≤ y, then z → x ≤ z → y and z x ≤ z y, 1 → x = x, 1 x = x, ((y → x ) x ) → x = y → x, ((y x ) → x ) x = y x, x → y ≤ (y → x ) 1, x y ≤ (y x ) → 1, (x → y) → 1 = (x → 1) (y → 1),(x y) 1 = (x 1) → (y → 1) x → 1 = x 1.
Deﬁnition 6 ([10,24]). A pseudoBCI algebra A is said to be an antigrouped pseudoBCI algebra if it satisﬁes the following identities: f or any x ∈ A, ( x → 1) → 1 = x or ( x 1) 1 = x. Proposition 3 ([24]). A pseudoBCI algebra A is antigrouped if and only if it satisﬁes: (G1) for all x, y, z ∈ A, (x → y) → (x → z) = y → z and (G2) for all x, y, z ∈ A, (x y) (x z) = y z. Proposition 4 ([24]). Let A = (A; ≤, →, , 1) be an antigrouped pseudoBCI algebra. Deﬁne Φ(A) = (A; +, −, 1) by x + y = ( x → 1) → y = (y 1) x, ∀ x, y ∈ A;
− x = x → 1 = x 1, ∀ x ∈ A. Then Φ(A) is a group. Conversely, let G = (G; +, −, 1) be a group. Deﬁne Ψ(G) = (G; ≤, →, , 1), where x → y = (− x ) + y, x y = y + (− x ), ∀ x, y ∈ G; x ≤ y i f and only i f (− x ) + y = 1 (or y + (− x ) = 1), ∀ x, y ∈ G. Then,Ψ(G) is an antigrouped pseudoBCI algebra. Moreover, the mapping Φ and Ψ are mutually inverse.
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Deﬁnition 7 ([27]). Let (A; ≤, →, , 1) be a pseudoBCI algebra. Denote K ( A ) = { x ∈ A  x ≤ 1}; AG ( A) = { x ∈ A( x → 1) → 1 = x }. We say that K(A) is the pseudoBCK part of A and AG ( A) is the antigrouped part of A. Deﬁnition 8 ([28]). A pseudoBCI algebra A is said to be a Ttype if it satisﬁes the following identities: (T1) for all x ∈ A, (x → 1 ) → 1 = x → 1 , or (x 1 ) 1 = x 1. Proposition 5 ([28]). A pseudoBCI algebra A is Ttype if and only if it satisﬁes: (T2) for all x ∈ A, x → (x → 1 ) = 1, or x (x 1 ) = 1. 3. Some New Concepts and Results By the deﬁnition of pseudoBCI/BCK algebra, we know that the direct product of two pseudoBCI/BCK algebras is a pseudoBCI/BCK algebra. That is, we have the following lemma. Lemma 1 ([20]). Let (X; → X , X , 1X ) and (Y; → Y , Y , 1Y ) be two pseudoBCI algebras. Deﬁne two binary operators →, on X × Y as follwos: for any (x1 , y1 ), (x2 , y2 ) ∈ X × Y,
( x1 , y1 ) → ( x2 , y2 ) = ( x1 →
X
x2 , y1 →
Y y2 );
( x1 , y1 ) ( x2 , y2 ) = ( x1 X x2 y1 Y y2 ); and denote 1 = (1X , 1Y ). Then (X × Y; →, , 1) is a pseudoBCI algebra. By the results in [18,20], we can easy to verify that the following lemma (the proof is omitted). Lemma 2. Let (K; →, , 1) be a pseudoBCK algebra, (G; →, , 1) an antigrouped pseudoBCI algebra and K∩G = {1}. Denote A = KG b and deﬁne the operations →, on A as follows:
x→y=
xy=
⎧ ⎪ ⎪ ⎨x → y y
⎪ ⎪ ⎩x → 1 ⎧ ⎪ ⎪ ⎨x y y
⎪ ⎪ ⎩x 1
i f x, y ∈ K or x, y ∈ G i f x ∈ K,
y∈G
i f y ∈ K {1}, x ∈ G i f x, y ∈ K or x, y ∈ G i f x ∈ K,
y∈G
i f y ∈ K {1}, x ∈ G
Then (A; →, , 1) is a pseudoBCI algebra. Deﬁnition 9. Let K be a pseudoBCK algebra and G be an antigrouped pseudoBCI algebra, K∩G = {1}. If the operators →, are deﬁned on A = K∪G according to Lemma 2, then (A; →, , 1) is a pseudoBCI algebra, we call A to be a KGunion of K and G and denote by A = K⊕KG G. Deﬁnition 10. Let ( X, ≤) is a partial ordered set with 1 as a constant element. For x in X, we call x a quasimaximal element of X, if for any a ∈ X, x ≤ a ⇒ x = a or a = 1. Deﬁnition 11. Let (G,*) be a grouoid, x ∈ G. Then x is called a quasileft unit element of G, if it satisﬁes:
∀y ∈ G, x ∗ y = y when x = y.
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Theorem 1. Let (A; ≤, →, , 1) be pseudoBCK algebra. Then the following conditions are equivalent: (a1) (a2) (a3) (a4) (a5)
∀ x ∈ A, x is a quasimaximal element; ∀ x ∈ A, y ∈ A − {1}, x ≤ y implies x = y; ∀ x ∈ A, x is a quasileft unit elemen w.r.t →, , that is, x = y implies x → y = y and x y = y ; ∀ x, y ∈ A, x = y implies x → y = y ; ∀ x, y ∈ A, x = y implies x y = y .
Proof. (a1) ⇒ (a2) : Suppose that x ∈ A, y ∈ A − {1} and x ≤ y. Case 1: If x = 1, it is follows that 1 = x ≤ y ≤ 1, that is, x = y = 1. Case 2: If x = 1, by (a1) and Deﬁnition 10, from x ≤ y and y = 1, we have x = y. Therefore, (a2) hold. (a2) ⇒ (a3) : For any x, y in A, by Proposition 1 (6) and Deﬁnition 4 (2), we have x ≤ y → x , y ≤ x → y, x ≤ ( x → y) y . Assume x = y. If y → x = 1 , then x → y = 1 (since, if x → y = 1 , then form y → x = 1 and x → y = 1 we get x = y, this is contradictory to the hypothesis x = y). Thus, from y ≤ x → y and x → y = 1, using (a2) we have y = x → y . If y → x = 1 , from this and x ≤ y → x and applying (a2), we have x = y → x . Thus, (i) (ii)
when ( x → y) y = 1 , we can get x → y ≤ y ≤ x → y , that is, y = x → y ; when ( x → y) y = 1 , from this and x ≤ ( x → y) y, using (a2) we have x = ( x → y) y. Combine the aforementioned conclusion x = y → x , we can get x = y → x = y → (( x → y) y) = ( x → y) (y → y) = ( x → y) 1 = 1,
It follows that y = 1 → y = x → y . Therefore, based on the above cases we know that x = y implies y = x → y . Similarly, we can prove that x = y implies y = x y . (a3) ⇒ (a4): Obviously. (a4) ⇒ (a5): Suppose x = y. Applying (a4), x → y = y. Also, by Definition 4 (2), x ≤ (x y) → y , thus x → [( x y) → y] = 1 . Case 1: If x = ( x y) → y , using (a4), x → [( x y) → y] = [( x y) → y]. Hence, ( x y) → y = 1. Moreover, y → ( x y) = x (y → y) = x 1 = 1. Therefore, y = x y. Case 2: If x = ( x y) → y , then x y = y . In fact, if x y = y , using (a4), ( x y) → y = y, it follows that x = y, this is a contradiction with x = y. By above results we know that (a5) hold. (a5) ⇒ (a1): Assume that x ∈ X, a ∈ X and x ≤ a. Then x a = 1 . If x = a, by (a5), x a = a , then a = x a = 1 . This means that x ≤ a implies x = a or a = 1. By Theorem 1 and Deﬁnition 3 we get Corollary 1. Let (A; ≤, →, , 1) be a pseudoBCK algebra. Then every element of A is quasimaximal if and only if A is a quasialternating BCKalgebra.
4. The Class of PseudoBCI Algebras in Which Every Element is QuasiMaximal Example 1. Let A = {a, b, c, d, e, f, g, 1}. Deﬁne operations → and on A as following Cayley Tables 1 and 2. Then A is pseudoBCI algebra in which every element is quasimaximal. 310
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Table 1. The Cayley table of operation →.
→ a b c d e f g 1
a b c
d e
f
g
1
1 a c d f e g a
d d f 1 c g d d
f f d g e 1 f f
g g e f d c 1 g
1 1 c d f e g 1
b 1 c d f e g b
c c 1 e g d c c
e e g c 1 f e e
Table 2. The Cayley table of operation .
a b c
d e
f
g
1
a b c d e f g 1
1 a c d f e g a
d d f 1 c g d d
f f d g e 1 f f
g g e f d c 1 g
1 1 c d f e g 1
b 1 c d f e g b
c c 1 e g d c c
e e g c 1 f e e
Deﬁnition 12. A pseudoBCI/BCK algebra A is said to be a QMpseudoBCI/BCK algebra if every element of A is quasimaximal. Theorem 2. Let (A; ≤, →, , 1) be a pseudoBCI algebra. Then A is a QMpseudoBCI algebra if and only if it satisﬁes: f or any x, y ∈ A − {1}, x ≤ y ⇒ x = y. Proof. If A is a QMpseudoBCI algebra, by Deﬁnitions 10 and 12, the above condition is satisﬁed. Conversely, assume that x, y ∈ A, x ≤ y. If x = 1, then 1 = x ≤ y, it follows that x = y = 1, by Proposition 2 (1). If x = 1, y = 1, then x = y by the condition. This means that x is a quasi maximal element in A, hence, A is a QMpseudoBCI algebra. By Theorem 1 we know that a pseudoBCK algebra is a QMpseudoBCK algebra if and only if it is a quasialternating BCKalgebra. It will be proved that any QMpseudoBCI algebra is constructed by the combination of a quasialternating BCKalgebra and an antigrouped pseudo BCI algebra (a grouplike algebra). Lemma 3 ([27]). Let A be a pseudoBCI algebra, K(A) the pseudoBCK part of A. If AG(A) = (A − K(A))∪{1} is subalgebra of A, then (∀ x, y ∈ A) (1) (2)
If x ∈ K ( A) and y ∈ A − K ( A), then x → y = x y = y. If x ∈ A − K ( A) and y ∈ K ( A), then x → y = x y = x → 1.
Applying the results in [24,27] we can easy to verify that the following lemma is true (the proof is omitted). Lemma 4. Let A be an antigrouped pseudoBCI algebra. Then (1)
for any x, y in A, x ≤ y implies x = y; 311
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(2)
for any x, y in A, x = ( x → y) y = ( x y) → y.
Theorem 3. Let A be a pseudoBCI algebra, K(A) the pseudoBCK part of A and AG(A) the antigrouped part of A. The following statements are equivalent: (1) (2) (3) (4) (5) (6)
A is a QMpseudoBCI algebra; K(A) is quasialternating BCKalgebras and AG(A) = (A − K(A)) ∪ {1}; ∀ x, y ∈ A, x = y implies ( x → y) y = ( x → 1) → 1; ∀ x, y ∈ A, x = y implies ( x y) → y = ( x → 1) → 1; ∀ x, y ∈ A, x = y implies ( x → y) → y = ( x → 1) → 1; ∀ x, y ∈ A, x = y implies ( x y) y = ( x → 1) → 1.
Proof. (1) ⇒ (2): Suppose that A is a QMpseudoBCI algebra. Then, for all x, y ∈ K ( A), by Corollary 1, we can know K(A) is quasialternating BCKalgebras. If x ∈ A − K ( A), then x → 1 = 1 and ( x → 1) → 1 = 1. Since x ≤ ( x → 1) → 1 , by Deﬁnition 12 we have x = ( x → 1) → 1 . Thus, (A − K(A))∪{1} ⊆ AG(A). On the other hand, obviously, AG(A) ⊆ (A − K(A))∪{1}. Hence AG(A) = (A − K(A))∪{1}. (2) ⇒ (3): Assume that (2) hold. For any x, y in A, x = y, Case 1: x, y ∈ K ( A). Then x → 1 = y → 1 = 1. Because K(A) is quasialternating BCKalgebra, using Theorem 1, x → y = y . Thus
( x → y) y = y y = 1 = 1 → 1 = ( x → 1) → 1. Case 2: x, y ∈ AG ( A). Since AG(A) is an antigrouped pseudoBCI subalgebra of A, then by Lemma 4 we get ( x → y) y = x = ( x → 1) → 1. Case 3: x ∈ K ( A), y ∈ AG ( A). Then x → 1 = 1. Applying Lemma 3 (1), x → y = y. Then
( x → y) y = y y = 1 = 1 → 1 = ( x → 1) → 1. Case 4: x ∈ AG( A), y ∈ K( A). Then x = (x → 1) → 1, y → 1 = 1. Applying Lemma 3 (2), x → y = x → 1. When x = 1, then (x → y) y = (x → 1) → 1; when x = 1, then x → 1 ∈ A − K( A), using Lemma 3 (2), ( x → 1) y = ( x → 1) → 1 Hence,
( x → y) y = ( x → 1) y = ( x → 1) → 1. (3) ⇒ (1): Assume that x ≤ y and x = y. We will prove that y = 1. By (3), we have y = 1 y = ( x → y) y = ( x → 1) → 1. Case 1: when x ∈ K ( A), then x → 1 = 1 , so y = 1. Case 2: when x ∈ X − K ( A), then ( x → 1) → 1 = x , so y = x, this is a contradiction with x = y. Therefore, for all x ∈ A, x is a quasimaximal element of A. (4) ⇒ (2): Suppose (4) hold. For any x, y in A. If x, y ∈ K ( A), x = y, by (4),
( x y) → y = ( x → 1) → 1 = 1.
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Then, x y ≤ y. Since K(A) is a pseudoBCK subalgebra of A, using Proposition 1 (6), y ≤ x y . It follows that y ≤ x y ≤ y , that is, x y = y . Thus, applying Theorem 1, K(A) is a quasialternating BCKalgebra. If x ∈ A − K ( A), we prove that ( x → 1) → 1 = x . Assume ( x → 1) → 1 = x, by (4), we have
{[( x → 1) → 1] x } → x = {[( x → 1) → 1] → 1} → 1. Using Proposition 2 (9) and (12),
{[( x → 1) → 1] → 1} → 1 = ( x → 1) → 1. Thus
{[( x → 1) → 1] x } → x = ( x → 1) → 1. Moreover, applying Proposition 2 (9), (11) and (12) we have
{[( x → 1) → 1] x } → 1 = {[( x → 1) → 1] 1} → ( x 1) = {[( x → 1) 1] → 1} → ( x 1) = ( x → 1) → ( x 1) = 1. This means that (( x → 1) → 1) → x ∈ K ( A). By Lemma 3 (1),
{[( x → 1) → 1] x } → x = x. Hence, (x → 1) → 1 = x . This is contraction with (x → 1) → 1 = x. Therefore, (x → 1) → 1 = x and x ∈ AG ( A). It follows that (A − K(A))∪{1} ⊆ AG(A). Obviously, AG(A) ⊆ (A − K(A))∪{1}. So AG(A) = (A − K(A))∪{1}. (2) ⇒ (4): It is similar to (2) ⇒ (3). It follows that (4) ⇔ (2). Similarly, we can prove (5) ⇔ (2), (6) ⇔ (2). Theorem 4. Let (A; ≤, →, , 1) be a pseudoBCI algebra, AG(A) the antigrouped part of A, K(A) the pseudoBCK part of A. Then A is a QMpseudoBCI algebra if and only if K(A) is a quasialternating BCKalgebra and A = K(A)⊕KG AG(A). Proof. If A is a QMpseudoBCI algebra, then K(A) is a quasialternating BCKalgebra and A = K(A)⊕KG AG(A), by Lemma 3 and Theorem 3. Conversely, if K(A) is a quasialternating BCKalgebra, then every element in K(A) is quasimaximal; if A = K(A)⊕KG AG(A), then AG(A) = (A − K(A))∪{1}, it follows that every element in A − K(A) is quasimaximal. By Deﬁnition 12, we know that A is a QMpseudoBCI algebra. 5. Weak Associative PseudoBCI Algebras Deﬁnition 13. A pseudoBCI/BCK algebra A is said to be weak associative, if it satisﬁes: f or any, y, z ∈ A, ( x → y) → z = x → (y → z) when ( x = y, x = z). Example 2 Let A = {a, b, c, d, e, f, 1}. Deﬁne operation → on A as following Cayley Table 3. Then A is a weak associative pseudoBCI algebra, where = → .
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Table 3. The Cayley table of the operation →.
→
a
b
c
d
e
f
1
a b c d e f 1
1 a a d e f a
b 1 b d e f b
c c 1 d e f c
d d d 1 f e d
e e e f 1 d e
f f f e d 1 f
1 1 1 d e f 1
Theorem 5. Let (A; ≤, →, , 1) be a weak associative pseudoBCI algebra. Then A is a QMpseudoBCI algebra and a Ttype pseudoBCI algebra. Proof. For any x, y in A, x = y, then (by Deﬁnition 13)
( x → y) → y = x → (y → y) = x → 1. Thus, if x = 1, then ( x → 1) → 1 = x → 1. Obviously, when x = 1, ( x → 1) → 1 = x → 1. Hence, from Deﬁnition 13 we get that for any x, y in A, x = y ⇒ ( x → y) → y = ( x → 1) → 1. Applying Theorem 3 (5) we know that A is a QMpseudoBCI algebra. Moreover, we already prove that ( x → 1) → 1 = x → 1 for any x in A, by Deﬁnition 8 we know that A is a Ttype pseudoBCI algebra. The inverse of Theorem 5 is not true. Since (d → c ) → c = d → 1, so the QMpseudoBCI algebra in Example 1 is not weak associative. The following example shows that a Ttype pseudoBCI algebra may be not a QMpseudoBCI algebra. Example 3. Let A = {a, b, c, d, 1}. Deﬁne operations → and on A as following Cayley Tables 4 and 5. Then A is a Ttype pseudoBCI algebra but it is not a QMpseudoBCI algebra, since
( b → c ) → a = a = 1 = b → ( c → a ). Table 4. The operation → in the Ttype pseudoBCI algebra.
→
a
b
c
d
1
a b c d 1
1 b b d a
1 1 b d b
1 1 1 d c
d d d 1 d
1 1 1 d 1
Table 5. The operation in the Ttype pseudoBCI algebra.
a
b
c
d
1
a b c d 1
1 c a d a
1 1 b d b
1 1 1 d c
d d d 1 d
1 1 1 d 1
Lemma 5 ([16,24]). Let (A; →, 1) be a BCIalgebra. Then the following statements are equivalent: 314
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(1) (2) (3)
A is associative, that is, ( x → y) → z = x → (y → z) for any x, y, z in A; for any x in A, x → 1 = x; for all x, y in A, x → y = y → x.
Theorem 6. Let (A; ≤, →, , 1) be a weak associative pseudoBCI algebra, AG(A) the antigrouped part of A, K(A) the pseudoBCK part of A. Then (1) (2) (3) (4)
K(A) is quasialternating BCKalgebra and AG(A) = (A − K(A))∪{1}; For any x in AG(A), x → 1 = x 1 = x ; For any x, y in A, x → y = x y, that is, A is a BCIalgebra; AG(A) is an Abel group, that is, AG(A) is associative BCIalgebra.
Proof. (1) It follows from Theorems 5 and 3. (2) For any x in AG(A), then ( x → 1) → 1 = x. We will prove that x → 1 = x. If x = 1, obviously, x → 1 = x. If x = 1, then ( x → 1) → 1 = x → 1 by Deﬁnition 13. Thus, x → 1 = ( x → 1) → 1 = x. Applying Proposition 2 (12) we have x 1 = x → 1 = x. (3) For any x, y in A, (i) (ii)
when x, y in K(A), by (1), K(A) is a BCKalgebra, so x → y = x y; when x, y in (A − K(A)), by (1) and (2), applying Proposition 2 (11), x → y = ( x → y) → 1 = ( x → 1) (y → 1) = x y;
(iii) when x in K(A), y in (A − K(A)), using Lemma 3 (1), x → y = x y; (iv) when y in K(A), x in (A − K(A)), using Lemma 3 (2), x → y = x y; Therefore, for all x, y in A, x → y = x y. It follows that A is a BCIalgebra. (4) Applying (2), by Lemma 5 we know that AG(A) is an Abel group, that is, AG(A) is associative BCIalgebra. From Theorems 6 and 4 we immediately get Theorem 7. Let (A; ≤, →, , 1) be a pseudoBCI algebra, AG(A) the antigrouped part of A, K(A) the pseudoBCK part of A. Then A is a weak associative pseudoBCI algebra if and only if K(A) is a quasialternating BCKalgebra, AG(A) is an Abelian group and A = K(A)⊕KG AG(A). Theorem 8. Let (A; ≤, →, , 1) be a pseudoBCI algebra. Then the following conditions are equivalent: (1) (2) (3) (4)
for any x, for any x, for any x, for any x,
y, z y, z y, z y, z
∈ ∈ ∈ ∈
A, A, A, A,
( x → y) → z = x → (y → z) when ( x = y, ( x y) z = x (y z) when ( x = y, ( x → y) z = x → (y z) when ( x = y, ( x y) → z = x (y → z) when ( x = y,
x = z ); x = z ); x = z ); x = z ).
Proof. (1) ⇒ (2) : It follows from Deﬁnition 13 and Theorem 6. (2) ⇒ (1) : Similar to the discussion process from Deﬁnition 13 to Theorem 6, we can obtain a result similar to Theorem 6. That is, from (2) we can get that A is a BCIalgebra. Hence, (2) implies (1). 315
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Similarly, (3) ⇔ (1) and (4) ⇔ (1). Finally, we discuss the relationships among general pseudoBCI algebras, QMpseudoBCI algebras and weak associative pseudoBCI algebras (WApseudoBCI algebras). In fact, in every Ttype pseudoBCI algebra, there is a maximal WApseudoBCI subalgebra. That is, if (A; ≤, →, , 1) is a Ttype pseudoBCI algebra, AG(A) the antigrouped part of A, K(A) the pseudoBCK part of A, then Kqm (A)∪AG(A) is a WApseudoBCI subalgebra of A, where Kqm (A) is the set of all quasimaximal element in K(A). For example, {c, d, 1} is a WApseudoBCI subalgebra of the pseudoBCI algebra A in Example 3. In general, in every pseudoBCI algebra, there is a maximal QMpseudoBCI subalgebra. That is, if (A; ≤, →, , 1) is a pseudoBCI algebra, AG(A) the antigrouped part of A, K(A) the pseudoBCK part of A, then Kqm (A)∪AG(A) is a QMpseudoBCI subalgebra of A, where Kqm (A) is the set of all quasimaximal element in K(A). 6. Conclusions In the study of pseudoBCI algebras, the structures of various special pseudoBCI algebras are naturally an important problem. At present, the structures of several subclasses such as quasialternating pseudoBCI algebras and antigrouped pseudoBCI algebras are clear. In this paper, we have studied an important subclass of pseudoBCI algebras, that is, QMpseudoBCI algebras in which every element is quasimaximal. We obtain a very clear structure theorem of this subclass. At the same time, we have studied a class of more special pseudoBCI algebras, that is, weak associative (WA) pseudoBCI algebras in which every element is weak associative and obtained the structure theorem of this subclass. These results enrich the research content of pseudoBCI algebras and clearly presented the relationships between various subclasses, which can be illustrated as Figure 1. Finally, we show that the two types of pseudoBCI algebras are very important, since (1) every pseudoBCI algebra contains a subalgebra which is QMpseudoBCI algebra, (2) every Ttype pseudoBCI algebra contains a subalgebra which is WApseudoBCI algebra. As a further study direction, we will discuss the integration of related topics in the light of some new research ﬁndings in [32–34].
Figure 1. Main results in this paper.
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Author Contributions: The contributions of the authors are roughly equal. X.Z. and X.W. initiated the research; X.W. wrote the draft and X.Z. completed ﬁnal version. Funding: This research was funded by National Natural Science Foundation of China grant number 61573240. Conﬂicts of Interest: The authors declare no conﬂict of interest.
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Xin, X.L.; Li, Y.J.; Fu, Y.L. States on pseudoBCI algebras. Eur. J. Pure Appl. Math. 2017, 10, 455–472. Zhang, X.H.; Park, C.; Wu, S.P. Soft set theoretical approach to pseudoBCI algebras. J. Intell. Fuzzy Syst. 2018, 34, 559–568. Zhang, X.H.; Smarandache, F.; Liang, X.L. Neutrosophic duplet semigroup and cancellable neutrosophic triplet groups. Symmetry 2017, 9, 275. [CrossRef] Zhang, X.H.; Bo, C.X.; Smarandache, F.; Park, C. New operations of totally dependent neutrosophic sets and totally dependentneutrosophic soft sets. Symmetry 2018, 10, 187. [CrossRef] Zhang, X.H.; Bo, C.X.; Smarandache, F.; Dai, J.H. New inclusion relation of neutrosophic sets with applications and related lattice structure. Int. J. Mach. Learn. Cyber. 2018, 9, 1753–1763. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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SS symmetry Article
A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons Qing Yang 1 , Zengtai You 2 and Xinshang You 3, * 1 2 3
*
Faculty of Accounting, Shanxi University of Finance and Economics, Taiyuan 030006, China; [email protected] College of Computer Science and Engineering, Dalian Minzu University, Dalian 116600, China; [email protected] College of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, China Correspondence: [email protected]
Received: 6 September 2018; Accepted: 26 September 2018; Published: 29 September 2018
Abstract: Let P be a planar point set with no three points collinear, k points of P be a khole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the smallest integer such that any planar points set containing at least 13 points with no three points collinear, contains a 3hole, a 4hole and a 5hole which are pairwise disjoint. Keywords: planar point set; convex polygon; disjoint holes
1. Introduction In this paper, we deal with the ﬁnite planar point set P in general position, that is to say, no three points in P are collinear. In 1935, Erd˝os and Szekeres [1], posed a famous combinational geometry question: Whether for every positive integer m ≥ 3, there exists a smallest integer ES(m), such that any set of n points (n ≥ ES(m)), contains a subset of m points which are the vertices of a convex polygon. It is a long standing open problem to evaluate the exact value of ES(m). Erd˝os and Szekeres [2] showed that ES(m) ≥ 2m−2 + 1, which is also conjectured to be sharp. We have known that ES(4) = 5 and ES(5) = 9. Then by using computer, Szekeres and Peters [3] proved that ES(6) = 17. The value of ES(m) for all m > 6 is unknown. For a planar point set P, let k points of P be a khole of P if the k points are the vertices of a convex polygon whose interior contains no points of P. Erd˝os posed another famous question in 1978. He asked whether for every positive integer k, there exists a smallest integer H (k), such that any set of at least H (k) points in the plane, contains a khole. It is obvious that H (3) = 3. Esther Klein showed H (4) = 5. Harborth [4] determined H (5) = 10, and also gave the conﬁguration of nine points with no empty convex pentagons. Horton [5] showed that it was possible to construct arbitrarily large set of points without a 7hole, That is to say H (k) does not exist for k ≥ 7. The existence of H (6) had been proved by Gerken [6] and Nicolás [7], independently. In [8], Urabe ﬁrst studied the disjoint holes problems when hewas considering the question about partitioning of planar point sets. Let Ch( P) stand for the convex hull of a point set P. A family of holes { Hi }i∈ I is called pairwise disjoint if Ch( Hi ) ∩ Ch( Hj ) = ∅, i = j; i ∈ I, j ∈ I. These holes are disjoint with each other. Determine the smallest integer n(k1 , ..., k l ), k1 ≤ k2 ≤ ... ≤ k l , such that any set of at least n(k1 , ..., k l ) points of the plane, contains a k i hole for every i, 1 ≤ i ≤ l, where the holes are disjoint. From [9], we know n(2, 4) = 6, n(3, 3) = 6. Urabe [8] showed that n(3, 4) = 7, while Hosono and Urabe [10] showed that n(4, 4) = 9. In [11], Hosono and Urabe also gave n(3, 5) = 10, 12 ≤ n(4, 5) ≤ 14 and 16 ≤ n(5, 5) ≤ 20.
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The result n(3, 4) = 7 and n(4, 5) ≤ 14 were reauthentication by Wu and Ding [12]. Hosono and Urabe [9] proved n(4, 5) ≤ 13. n(4, 5) = 12 by Bhattacharya and Das was published in [13], who also discussed the convex polygons and pseudotriangles [14]. Hosono and Urabe also changed the lower bound on H (5, 5) to 17 [9], and Bhattacharya and Das showed the upper bound on n(5, 5) to 19 [15]. Recently, more detailed discussions about two holes are published in [16]. Hosono and Urabe in [9] showed n(2, 3, 4) = 9, n(2, 3, 5) = 11, n(4, 4, 4) = 16. We showed n(3, 3, 5) = 12 in [17]. We have proved that n(3, 3, 5) = 12 [17], n(4, 4, 5) ≤ 16 [18] and also discuss a disjoint holes problem in preference [19]. In this paper, we will continue discussing this problem and prove that n(3, 4, 5) = 13. 2. Deﬁnitions The vertices are on convex hull of the given points,from the remaining interior points. Let V ( P) denote a set of the vertices and I ( P) be a set of the interior points of P.  P stands for the number of points contained in P. Let p1 , p2 , ..., pk be k points of P, we know that p1 , p2 , ..., pk be a khole H when the k points are the vertices of a convex polygon whose interior does not contain any point of P. And we simply say H = ( p1 p2 ...pk )k . As in [9], let l ( a, b) be the line passing points a and b. Determine the ¯ ab), respectively. closed halfplane with l ( a, b), who contains c or does not contain c by H (c; ab) or H (c; R is a region in the plane. An interior point of R is an element of a given point set P in its interior, and we say R is empty when R contains no interior points, and simply R = ∅. The interior region of the angular domain determined by the points a, b and c is a convex cone. It is denoted by γ( a; b, c). a is the apex. b and c are on the boundary of the angular domain. If γ( a; b, c) is not empty, we deﬁne an interior point of γ( a; b, c) be attack point α( a; b, c), such that γ( a; b, α( a; b, c)) is empty, as shown in Figure 1.
Figure 1. Figure of attack point.
For β = b or β = c of γ( a; b, c), let β be a point such that a is on the line segment ββ . γ( a; b , c) means that a lies on the segment bb . Let v1 , v2 , v3 , v4 ∈ P and (v1 v2 v3 v4 )4 be a 4hole, as shown in Figure 2. We name l (v3 , v4 ) a separating line, denoted by SL(v3 , v4 ) or SL4 for simple, when all of the remaining points of P locate in H (v1 ; v3 v4 ).
Figure 2. Figure of separating line.
We identify indices modulo t, when indexing a set of t points.
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3. Main Result and Proof Theorem 1. [9] For any planar point set with at least 13 points in general position, if there exists a separating line SL4 , which separates a 4hole from all of the remaining points, we always can ﬁnd a 3hole, a 4hole and a 5hole which are pairwise disjoint. From [20], we know that 13 ≤ n(3, 4, 5) ≤ 14. In this note we will give the exact value of n(3, 4, 5), that is the following theorem. Theorem 2. n(3, 4, 5) = 13, that is to say, 13 is the smallest integer such that any planar point set with at least 13 points in general position, we always can ﬁnd a 3hole, a 4hole and a 5hole which are pairwise disjoint. Proof. Let P be a 13 points set. CH ( P) = {v1 , v2 , ..., vl }. If we can ﬁnd a 5hole and a disjoint convex region with at least 7 points remained, we are done by n(3, 4) = 7 [8]. That is to say, if we ﬁnd a straight line which separates a 5hole from at least 7 points remained, the result is correct. We call such a line a cutting line through two points u and v in P, denoted by L5 (u, v). If we can ﬁnd a 4hole and the vertices number of the remaining points is more than 4, we are done by Theorem 1, where the two parts are disjoint. That is to say, if we can ﬁnd such a cutting line through two points m and n in P, denoted by L4 (m, n), our conclusion is correct. Therefore, in the following proof, if we can ﬁnd a cutting line L5 (u, v) or L4 (m, n), our conclusion must be true. In the following, we will assume there does not exist a separating line SL4 . Then there must exist a point pi , such that γ( pi ; vi , vi−1 ) and γ( pi ; vi−1 , vi ) are empty, as shown in Figure 3. Considering the 13 points, it is easy to know the conclusion is obvious right when V ( P) ≥ 7. Next, we discuss the considerations that 3 ≤ V ( P) ≤ 6.
Figure 3. Figure of point determined by two separating lines.
Case 1 V (P ) = 6. Let vi ∈ V (P ) for i = 1, 2, ...6. As shown in Figure 4, we have the points pi for i = 1, 2, ...6, such that the shaded region is empty and we have 1 point p7 remained.
Figure 4. Figure of V(P) = 6
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As the isomorphism of geometry from Figure 4, we only discuss one case. And the rest could be obtained in the same way. Assume γ(v1 ; p1 , v3 ) ∩ γ(v3 ; v1 , p2 ) = ∅. We have a cutting line L5 (v1 , α(v1 ; v3 , v6 )). Assume γ(v1 ; p1 , v3 ) ∩ γ(v3 ; v1 , p2 ) = ∅. We have a cutting line L5 (v1 , p4 ). Case 2 V (P ) = 5. Let vi ∈ V (P ) for i = 1, 2, 3, 4, 5. We have 5 friend points pi for i = 1, 2, 3, 4, 5 as shown in Figure 5. Then we have 3 points r1 , r2 , r3 remained.
Figure 5. Figure of V(P) = 5.
Assume γ( p1 ; v1 , p3 ) ∩ γ( p2 ; v3 , v2 ) = ∅. We have a cutting line L5 ( p1 , α( p1 ; p3 , v2 )). Assume γ( p3 ; v3 , p5 ) ∩ γ( p4 ; v4 , v5 ) = ∅. We have a cutting line L5 ( p3 , α( p3 ; p5 , p1 )). Assume γ( p1 ; v1 , p3 ) ∩ γ( p2 ; v3 , v2 ) = ∅ and γ( p3 ; v3 , p5 ) ∩ γ( p4 ; v4 , v5 ) = ∅. Suppose γ( p1 ; v2 , p3 ) ∩ γ( p5 ; v5 , p3 ) = ∅. If γ( p1 ; v1 , p3 ) ∩ γ( p2 ; v3 , v2 ) has two of the remaining points say r1 , r2 , r3 ∈ γ( p5 ; p3 , v5 ), let r1 = α( p3 ; p1 , v4 ): and if r2 ∈ γ(r1 ; p2 , p3 ) = ∅, we have a cutting line L5 (r1 , p3 ); and if r2 ∈ γ(r1 ; p1 , p3 ), we have (v2 v3 p2 )3 , ( p1 r1 r2 p3 v1 )5 and a 4hole from the remaining points; and if r2 ∈ γ(r1 ; p2 , p1 ), we have a cutting line L5 ( p1 , r1 ). If γ( p5 ; p3 , v5 ) ∩ γ(v4 ; p3 , p4 ) has two of the remaining points, symmetrically, the conclusion is also right. Suppose γ( p1 ; v2 , p3 ) ∩ γ( p5 ; v5 , p3 ) = ∅. We may suppose r1 ∈ γ( p1 ; v1 , p3 ) ∩ γ( p2 ; v3 , v2 ), r2 ∈ γ( p1 ; v2 , p3 ) ∩ γ( p5 ; v5 , p3 ), r3 ∈ γ( p3 ; v4 , p5 ) ∩ γ( p4 ; v4 , v5 ). If γ(r2 ; p1 , p3 ) = ∅, we have (v2 v3 p2 )3 , ( p1 r1 p3 r2 v1 )5 and a 4hole from the remaining points. If γ(r2 ; p3 , p1 ) = ∅, we have (v2 v3 p2 )3 , (r2 p1 r1 p3 α(r2 ; p3 , p1 ))5 and a 4hole from the remaining points. If γ(r2 ; p1 , p3 ) = ∅ and γ(r2 ; p3 , p1 ) = ∅, we have (v4 v5 p4 )3 , (r3 p5 v1 r2 p3 )5 and a 4hole from the remaining points. Case 3 V (P ) = 4. Let vi ∈ V (P ) for i = 1, 2, 3, 4. We have 4 friend points pi for i = 1, 2, 3, 4. Then we have 5 points r1 , r2 , r3 , r4 , r5 remained as shown in Figure 6.
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Figure 6. Figure of V(P) = 4.
If γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) = ∅ or γ( p3 ; v3 , v4 ) ∩ H ( p3 ; p2 p4 ) = ∅, we have a cutting line L5 ( p4 , α( p4 ; p2 , v1 )) or L5 ( p4 , α( p4 ; p2 , v4 )). Then we will consider that γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) = ∅ and γ( p3 ; v3 , v4 ) ∩ H ( p3 ; p2 p4 ) = ∅. Assume one of the ﬁve points say r1 ∈ γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) and the remaining four say ri ∈ γ( p3 ; v3 , v4 ) ∩ H ( p3 ; p2 p4 ), i = 2, 3, 4, 5. (If γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) has four points and γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) has one point, symmetrically, the conclusion is also right). Let r2 = α( p4 ; p2 , v1 ). Suppose r1 ∈ γ( p1 ; v1 , p2 ) or r1 ∈ γ( p1 ; v2 , p4 ). We always have a cutting line L5 ( p2 , p4 ). Suppose r1 ∈ γ( p1 ; p4 , r2 ) ∩ H ( p1 ; p2 p4 )). We have (v1 v4 p4 )3 , ( p1 v2 p2 r2 r1 )5 and a 4hole from the remaining points. Suppose r1 ∈ γ( p1 ; p2 , r2 ) ∩ H ( p1 ; p2 p4 ). We have (v2 v3 p2 )3 , ( p1 v1 p4 r2 r1 )5 and a 4hole from the remaining points. Assume two of the ﬁve points, say r1 , r2 ∈ γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) and the remaining three say ri ∈ γ( p3 ; v3 , v4 ) ∩ H ( p3 ; p2 p4 ), i = 3, 4, 5. (If γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) has three points and γ( p1 ; v1 , v2 ) ∩ H ( p1 ; p2 p4 ) has two points, symmetrically, our conclusion is also right.) Suppose γ( p2 ; v1 , p4 ) = ∅. If γ( p2 ; v1 , p1 ) = ∅, let r1 = α( p2 ; v1 , p1 ), we have (r2 p1 v2 )3 , ( p4 v1 r1 p2 α( p2 ; p4 , v2 ))5 and a 4hole from the remaining points. If γ( p2 ; v1 , p1 ) = ∅, we have (r1 r2 v2 )3 , ( p4 v1 p1 p2 α( p2 ; p4 , v2 ))5 and a 4hole from the remaining points. Suppose γ( p2 ; v1 , p4 ) = ∅. Let r1 = α( p2 ; p4 , v1 ). If r2 ∈ γ(r1 ; p1 , p2 ), we have (v1 v4 p4 )3 , (r1 r2 p1 v2 p2 )5 and a 4hole from the remaining points. If r2 ∈ γ(r1 ; p1 , p4 ), we have (v2 p2 v3 )3 , (v1 p1 r2 r1 p4 )5 and a 4hole from the remaining points. If r2 ∈ γ(r1 ; p2 , p4 ), we have (v1 v2 p1 )3 , ( p4 r1 r2 p2 α( p2 ; p4 , v2 )5 and a 4hole from the remaining points. Case 4 V (P ) = 3. Let v1 , v2 , v3 ∈ V (P ). We have 3 friend points p1 , p2 , p3 and 7 points remained. As shown in Figure 7, denote γ( p1 ; v2 , p3 ) ∩ γ( p3 ; v3 , p1 ) = T1 , γ( p1 ; v1 , p2 ) ∩ γ( p2 ; v3 , p1 ) = T2 , γ( p2 ; v2 , p3 ) ∩ γ( p3 ; v1 , p1 ) = T3 . Without loss of generality, we assume  T3  ≥  T1  ≥  T2 . (1)
 T3  = 7.
We have a cutting line L5 ( p2 , α( p2 ; p3 , v2 ))). (2)
 T3  = 6.
Name the remaining one r1 . If r1 ∈ γ( p3 ; v3 , p1 ) or r1 ∈ γ( p2 ; v3 , p1 ), we have a cutting line L5 ( p2 , p3 ). If r1 ∈ γ( p3 ; p1 , p2 ) ∩ γ( p1 ; p2 , p3 ): and if γ(r1 ; p3 , p1 ) = ∅, we have a cutting line
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L5 (r1 , α(r1 ; p3 , p1 )); and if γ(r1 ; p3 , p1 ) = ∅, we have (v1 v3 p3 )3 , (r1 p1 v2 p2 α(r1 ; p2 , p1 ))5 and a 4hole from the remaining points. (3)
 T3  = 5.
Name the remaining two points r1 , r2 . Then we will discuss the region γ( p3 ; v1 , p1 ), as shown in Figure 8.
Figure 7. Figure of V(P) = 5.
Figure 8. Figure of T3  = 5
Assume γ( p3 ; v1 , p1 ) = ∅. (If γ( p1 ; p2 , v2 ) = ∅, by the similar reason our conclusion is also right.) Let r1 = α( p3 ; p1 , p2 ). Suppose r1 ∈ γ( p2 ; p1 , p3 ). If r2 ∈ γ(r1 ; p3 , p1 ), we have a cutting line L5 ( p3 , r2 ). If r2 ∈ γ( p2 ; r1 , p1 ): and if γ(r1 ; p3 , p1 ) = ∅, we have (r2 p2 v2 )3 , ( p3 v1 p1 r1 α(r1 ; p3 , p1 ))5 and a 4hole from the remaining points; and if γ(r1 ; p3 , p1 ) = ∅, we have (v1 v2 p1 )3 , ( p3 r1 r2 p2 α( p2 ; p3 , v3 ))5 and a 4hole from the remaining points. Suppose r1 ∈ γ( p2 ; p1 , v3 ). If r2 ∈ γ(r1 ; p3 , p1 ), we have a cutting line L5 ( p3 , r2 ). If r2 ∈ γ(r1 ; p1 , p3 ), we have (r1 v2 p2 )3 , ( p3 v1 p1 r1 α(r1 ; p3 , p2 ))5 and a 4hole from the remaining points. Assume γ( p3 ; v1 , p1 ) = ∅ and γ( p1 ; p2 , v2 ) = ∅. Then we suppose γ( p3 ; v1 , p1 ) has one point say r1 and γ( p1 ; p2 , v2 ) has one point say r2 . If γ(r1 ; p1 , p2 ) = ∅, we have a cutting line L5 ( p2 , r1 ). If γ(r1 ; p1 , p2 ) = ∅, we have a cutting line L5 (r1 , α(r1 ; p2 , p3 ). (4)
 T3  = 4.
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Name the remaining three points r1 , r2 , r3 . Then we will discuss the region γ( p3 ; p1 , v3 ), as shown in Figure 9.
Figure 9. Figure of T3  = 4
(a) (b)
(c)
Assume r1 , r2 , r3 ∈ γ( p3 ; p1 , v3 ). Let r1 = α( p3 ; p1 , v3 ). We have (v1 p2 p3 )3 , (r1 p1 v2 p2 p3 )5 and a 4hole from the remaining points. Assume two of ri , i = 1, 2, 3, say r1 , r2 ∈ γ( p3 ; p1 , v3 ). Suppose r3 ∈ γ( p2 ; p1 , p3 ) ∩ γ( p3 ; p1 , p2 ). If γ(r3 ; p1 , p2 ) = ∅: we have a 4hole from {r4 , r5 , r6 , r7 , v3 }, ( p1 v2 p2 r3 α(r3 ; p1 , p2 ))5 and a 3hole from the remaining points. If γ(r3 ; p1 , p2 ) = ∅, we have(r3 p1 v2 p2 α(r3 ; p2 , p1 ))5 , (r1 r2 v1 )3 and a 4hole from the remaining points. If γ(r3 ; p1 , p2 ) = ∅ and γ(r3 ; p1 , p2 ) = ∅, we have a cutting line L4 ( p2 , r3 ). Assume one of ri , i = 1, 2, 3, say r1 ∈ γ( p3 ; p1 , v3 ). Suppose γ(r3 ; p1 , v2 ) = ∅. We have a cutting line L5 ( p3 , r2 ).
(d)
Suppose γ( p3 ; p1 , v2 ) = ∅. Let r2 = α( p3 ; p1 , v2 ). If r2 ∈ γ( p1 ; v1 , p2 ), we have a cutting line L5 (r2 , α(r2 ; p3 , p2 )). Then we suppose r2 ∈ γ( p1 ; p2 , p3 ). If r1 ∈ γ(r2 ; p2 , p1 ): and if r3 ∈ γ(r2 ; p3 , p1 ), we have a cutting line L5 (r2 , r3 ); and if r3 ∈ γ(r2 ; p2 , p1 ), we have (v1 r1 p3 )3 , ( p1 v2 p2 r3 r2 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; p2 , v2 ), we have (v1 v3 p3 )3 , (r1 p1 v2 r3 r2 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; v2 , p3 ), we have a cutting line L5 (v2 , p3 ). If r1 ∈ γ(r2 ; p2 , p3 ): and if r3 ∈ γ(r2 ; p3 , p1 ), we have a cutting line L5 (r2 , α(r2 ; p3 , p1 )); and if r3 ∈ γ(r2 ; p2 , p1 ), we have (v1 r1 p3 )3 , (r2 p1 v2 p2 r3 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; r1 , p2 ), we have (v1 v2 p1 )3 , ( p3 r1 r2 r3 p2 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; v2 , r1 ), we have (v1 v3 p3 )3 , (r1 p1 v2 r3 r2 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; p3 , v2 ), we have (v1 r1 p1 )3 , ( p3 r2 r3 v2 p2 )5 and a 4hole from the remaining points. Assume γ( p3 ; p1 , v3 ) = ∅. By the same reason, we also assume γ( p1 ; p2 , v1 ) = ∅. Then we will discuss the region γ(v1 ; p1 , p2 ), as shown in Figure 10.
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Figure 10. Figure of T3  = 4 with shaded region nonempty.
(d1)
Suppose γ(v1 ; p1 , p2 ) = ∅. Let r1 = α( p1 ; p3 , p2 ) within ( p1 p2 p3 ).
If γ(r1 ; p1 , p3 ) = ∅, we have (v2 v3 p2 )3 , (r1 p3 v1 p1 α(r1 ; p1 , p3 ))5 and a 4hole from the remaining points.
(d2)
If γ(r1 ; p1 , p3 ) = ∅: and if γ(r1 ; p3 , p1 ) = ∅, we have (v2 v3 p2 )3 , ( p3 v1 p1 r1 α(r1 ; p3 , p1 ))5 and a 4hole from the remaining points; and if γ(r1 ; p3 , p1 ) = ∅, let r2 = α(r1 ; p3 , p1 ) within ( p1 p2 p3 ), we have (v1 v3 p3 )3 , (r1 p1 v2 r2 r3 )5 and a 4hole from the remaining points when r3 ∈ γ(r2 ; r1 , v2 ) ∩ γ(r1 ; r2 , p1 ), we have (v1 p1 r1 p3 )4 , (r3 r2 v2 p2 α(r3 ; p2 , r2 ))5 and a 3hole from the remaining points when r3 ∈ γ(r2 ; p1 , v2 ) and γ( p3 ; p2 , v1 ) ∩ γ(r3 ; p2 , r2 ) = ∅, we have (v1 v2 p1 )3 , ( p3 r1 r2 r3 α(r3 ; p3 , r2 ))5 and a 4hole from the remaining points when r3 ∈ γ(r2 ; p1 , v2 ) and γ( p3 ; p2 , v1 ) ∩ γ(r3 ; p2 , r2 ) = ∅, we have (v1 r1 p3 )3 , ( p1 v2 p2 r3 r2 )5 and a 4hole from the remaining points when r3 ∈ γ(r2 ; p1 , p2 ), we have (v1 v2 p1 )3 , ( p3 r1 r2 r3 p2 )5 and a 4hole from the remaining points when r3 ∈ γ(r2 ; r1 , p2 ). Suppose γ(v1 ; p1 , p2 ) has one of the r1 , r2 , r3 , say r1 ∈ γ(r1 ; p1 , p2 ). Let r2 = α( p2 ; p1 , p3 ). If r2 ∈ γ(r1 ; p2 , p3 ), we have (v2 v3 p2 )3 , (r1 p1 v1 p3 r2 )5 and a 4hole from the remaining points. If r2 ∈ γ(r1 ; p1 , p3 ): and if r3 ∈ γ(r2 ; r1 , p3 ), we have (v1 v2 p1 )3 , (r3 r2 r1 p2 p3 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; p3 , p1 ), we have (v2 v3 p2 )3 , (v1 p1 r2 r3 p3 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; p1 , v1 ), we have a cutting line L5 (r1 , α(r1 ; p2 , p1 )) when γ(r1 ; p2 , p1 ) = ∅, we have (v2 v3 p2 )3 , (r3 r2 p1 r1 α(r3 ; r1 , r2 ))5 and a 4hole from the remaining points when γ(r1 ; p2 , p1 ) = ∅ and γ(r3 ; r1 , r2 ) = ∅, we have (r1 p1 v2 p2 )4 , ( p3 v1 r2 r3 α(r3 ; p3 , r2 ))5 and a 3hole from the remaining points when γ(r1 ; p2 , p1 ) = ∅ and γ(r3 ; r1 , r2 ) = ∅.
(d3)
If r2 ∈ γ(r1 ; p2 , p3 ), we have (v2 v3 p2 )3 , ( p3 v1 p1 r1 r2 )5 and a 4hole from the remaining points. Suppose γ(v1 ; p1 , p2 ) has two of the points r1 , r2 , r3 , say r1 , r2 ∈ γ(r1 ; p1 , p2 ). Let r1 = α( p2 ; p1 , p3 ).
If γ(r1 ; p2 , p1 ) = ∅, we have a cutting line L5 (r1 , α(r1 ; p2 , p1 )).
If γ(r1 ; p2 , p1 ) = ∅, let r2 = α( p1 ; p2 , p2 ): and if r2 ∈ γ(v1 ; p2 , p3 ), we have a cutting line L5 (r2 , r3 ) when r3 ∈ γ(r2 ; p1 , p3 ), we have (v1 p1 v2 )3 , ( p3 r2 r3 r1 p2 )5 and a 4hole from the remaining points when r3 ∈ γ(r2 ; p3 , r1 ), we have (v2 v3 p2 )3 , (r2 v1 p1 r1 p3 )5
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(d4)
and a 4hole from the remaining points when r3 ∈ γ(r1 ; r2 , p1 ), we have (v1 p3 v3 )3 , (r1 p1 v2 p2 r3 )5 and a 4hole from the remaining points when r3 ∈ γ(r1 ; p2 , p1 ); and if r2 ∈ γ(v1 ; p1 , p2 ), we have ( p1 r1 p2 v2 )4 , ( p1 v1 r1 r3 α(r3 ; p3 , r2 ))5 and a 3hole from the remaining points when γ(r2 ; r3 , p1 ) ∩ γ( p2 ; p3 , v2 ) = ∅, we have a cutting line L5 (r1 , α(r1 ; p2 , p1 )), when γ(r1 ; p2 , p1 ) = ∅, we have (v1 v3 p3 )3 , (r3 r2 p1 r1 α(r1 ; r3 , p1 ))5 and a 4hole from the remaining points when γ(r2 ; r3 , p1 ) ∩ γ( p2 ; p3 , v2 ) = ∅ and γ(r1 ; p2 , p1 ) = ∅. Suppose γ(v1 ; p1 , p2 ) has all of the three points r1 , r2 , r3 . Let r1 = α( p1 ; p3 , p2 ), r2 = α ( p1 ; p2 , p3 ).
If γ(r1 ; p3 , p1 ) = ∅ or γ(r2 ; p2 , p1 ) = ∅, we always have a cutting line L5 .
If γ(r1 ; p3 , p1 ) = ∅ and γ(r2 ; p2 , p1 ) = ∅: and if r3 ∈ γ(r1 ; p1 , p3 ), we have a cutting line L5 ( p3 , r1 ); and if r3 ∈ γ(r1 ; p3 , p2 ) ∩ γ(r2 ; p2 , r1 ), we have (v1 v2 p1 )3 , ( p3 r1 r3 r2 p2 )5 and a 4hole from the remaining points; and if r3 ∈ γ(r2 ; p1 , p2 ), we have a cutting line L5 ( p2 , r2 ); and if r3 ∈ γ(r2 ; p1 , v2 ), we have (v1 p1 r1 p3 )4 , (r3 r2 v2 p2 α(r3 ; p2 , r2 ))5 and a 3hole from the remaining points when γ(r3 ; p2 , r2 ) ∩ γ(v1 ; p2 , p3 ) = ∅, we have (v2 v3 p2 )3 , (r1 p1 r2 r3 α(r3 ; p3 , r2 ))5 and a 4hole from the remaining points when γ(r3 ; p2 , r2 ) ∩ γ(v1 ; p2 , p3 ) = ∅. (5) (a)
 T3  = 3. Let r1 , r2 , r3 ∈ T3 .  T1  = 3. Let r4 , r5 , r6 ∈ T1 . Name the remaining one point r7 . Assume r7 ∈ γ(v2 ; p3 , p2 ), as shown in Figure 11.
Figure 11. Figure of T1  = 3.
(b)
Symmetrically, our conclusion is also right when r7 ∈ γ(v2 ; p3 , p1 ). Let r4 = α( p3 ; p1 , v3 ). We have (r5 r6 v1 )3 , (r4 p1 v2 r7 p3 )5 and a 4hole from the remaining points.  T1  = 2. Let r4 , r5 ∈ T1 . Name the remaining two points r6 , r7 . (b1)
 T2  = 2. Let r6 , r7 ∈ T2 .
Assume γ(v1 ; p1 , p2 ) = ∅. Let r4 = α( p2 ; v1 , p3 ). Suppose r5 ∈ γ(r4 ; p2 , p3 ). We have a cutting line L5 ( p1 , p3 ). Suppose r5 ∈ γ(r4 ; v1 , p3 ). If γ(r5 ; p3 , r4 ) = ∅, we have a cutting line L5 (r5 , α(r5 ; p3 , v4 )). If γ(r5 ; p3 , r4 ) = ∅, we have a cutting line L5 (r1 , α(r1 ; p1 , p2 ) where r1 = α( p1 ; p3 , p2 ). Suppose r5 ∈ γ(r4 ; p2 , v1 ). We have (r6 r7 v2 )3 , (r4 v1 p1 p2 r5 )5 and a 4hole from the remaining points. 327
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Assume γ(v1 ; p1 , p2 ) has one of r4 , r5 . Let r4 ∈ α(v1 ; p1 , p2 ). Suppose r5 ∈ α(r4 ; p1 , v1 ). If γ(r4 ; p2 , v1 ) = ∅, we have (r2 r3 v3 )3 , (r5 r4 p2 r1 p3 )5 and a 4hole from the remaining points where r1 = α( p2 ; p3 , v3 ). If γ(r4 ; p2 , v1 ) = ∅, we have ( p1 v2 r7 )3 , (v1 r4 r6 p2 r5 )5 and a 4hole from the remaining points where r6 = α(r4 ; p2 , v1 ).
(b2)
Assume γ(v1 ; p1 , p2 ) has r4 , r5 . Let r4 ∈ α( p2 ; v1 , p1 ), r1 = α( p2 ; p3 , v3 ). we have (r2 r3 v3 )3 , ( p2 r1 p3 v1 r4 )5 and a 4hole from the remaining points.  T2  = 1. Let r6 ∈ T2 and r7 ∈ ( p1 p2 p3 ), as shown in Figure 12.
Figure 12. Figure of T2  = 1.
(b3)
Assume r6 ∈ γ(r7 ; p3 , p2 ). We have (r2 r3 v3 )3 , ( p3 r7 r6 p2 r1 )5 and a 4hole from the remaining points where r1 = α( p2 ; p3 , v3 ). Assume r6 ∈ γ(r7 ; p3 , v2 ). We have (r4 r5 v1 )3 , ( p1 v2 r6 r7 p3 )5 and a 4hole from the remaining points. Assume r6 ∈ γ(r7 ; p1 , v2 ). If γ(r7 ; r6 , p2 ) = ∅, we have (r2 r3 v3 )3 , ( p6 r7 r1 p2 v2 )5 and a 4hole from the remaining points where r1 = α(r7 ; p2 , r6 ). If γ(r7 ; r6 , p2 ) = ∅, we have (v2 v3 p2 )3 , ( p1 r6 r7 r1 p3 )5 and a 4hole from the remaining points where r1 = α(r7 ; p3 , p1 ).  T2  = 0. Let r6 , r7 ∈ ( p1 p2 p3 ). Then we will discuss the region γ( p3 ; p1 , v1 ), as shown in Figure 13.
Figure 13. Figure of T2  = 0.
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Assume γ( p3 ; p1 , v1 ) = ∅. Suppose γ(r6 ; p3 , v2 ) = ∅. We have ( p3 p1 v2 r6 α(r6 ; p3 , v2 ))5 , (r4 r5 v1 )3 and a 4hole from the remaining points. Suppose γ(r6 ; p3 , v2 ) = ∅. If γ(r7 ; r6 , p2 ) ∩ γ( p2 ; r7 , v2 ) = ∅, we have (r2 r3 v3 )3 , (r6 v2 p2 r1 r7 )5 and a 4hole from the remaining points where r1 = α(r7 ; p2 , r6 ). If γ(r7 ; r6 , p2 ) ∩ γ( p2 ; r7 , v2 ) = ∅, we have (v2 v3 p2 )3 , (r1 r7 r6 p1 α(r1 ; p1 , p3 ))5 and a 4hole from the remaining points where r1 = α( p1 ; r7 , p3 ) within γ( p3 ; p2 , v3 ). Assume γ( p3 ; p2 , v1 ) = ∅. We have (r2 r3 v3 )3 , (r1 p3 r6 v2 p2 )5 and a 4hole from the remaining points where r1 = α( p3 ; p2 , v3 ) and r6 = α( p3 ; v2 , p1 ). Assume γ( p3 ; p1 , v2 ) = ∅ and γ( p3 ; p1 , v2 ) = ∅. We may assume r6 ∈ γ( p3 ; p1 , v2 ) and r7 ∈ γ( p3 ; p1 , v2 ). Suppose r7 ∈ γ(r6 ; p2 , p1 ). We have (r4 r5 v1 )3 , (r6 r7 p2 v2 p1 )5 and a 4hole from the remaining points. Suppose r7 ∈ γ(r6 ; p3 , p1 ). If γ(r7 ; r6 , p2 ) = ∅, we have (r2 r3 v3 )3 , (r7 r6 v2 p2 r1 )5 and a 4hole from the remaining points where r1 = α( p2 ; r7 , v2 ). If γ(r7 ; r6 , p2 ) = ∅: and if γ(r7 ; p1 , v1 ) = ∅, we have (r2 v3 p2 )3 , (v1 p1 r6 r7 r4 )5 and a 4hole from the remaining points where r4 = α(r7 ; p1 , p3 ) within γ( p3 ; p1 , v1 ); and if γ(r7 ; p1 , v1 ) = ∅, we have (r2 v3 p2 )3 , (r4 p1 r6 r7 r1 )5 and a 4hole from the remaining points where r4 = α(r7 ; p1 , v1 ). (c)
 T1  = 1. Let r4 ∈ T1 . (c1)
 T2  = 1. Let r5 ∈ T2 and r6 , r7 ∈ ( p1 p2 p3 ). Firstly, consider r4 ∈ γ(v1 ; p1 , p2 ), then we will discuss the region γ(v1 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅, as shown in Figure 14.
Figure 14. Figure of T1  = 1 and T2  = 1.
Assume γ(v1 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅. We have a cutting line L5 (r4 , α(r4 ; p2 , p1 )). Assume γ(v1 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅. Let r6 = α( p2 ; p1 , v1 ). If γ(r6 ; p2 , p1 ) = ∅, we have a cutting line L5 (r4 , α(r6 ; p2 , p1 )). Then we may assume γ(r6 ; p2 , p1 ) = ∅.
Suppose r5 ∈ γ(r6 ; v2 , r4 ). If r7 ∈ γ(r6 ; p2 , r4 ), we have ( p1 v2 r5 )3 , (r4 r7 r6 p2 α(r4 ; p2 , p3 ))5 and a 4hole from the remaining points. If r7 ∈ γ(r6 ; p1 , r4 ), we have a cutting line L5 (r4 , r6 ).
Suppose r5 ∈ γ(r6 ; v2 , p1 ). If γ(r6 ; r5 , p1 ) = ∅, we have (v1 r4 p1 )3 , (r6 r5 v2 p2 α(r6 ; p2 , r5 ))5 and a 4hole from the remaining points. If γ(r6 ; r5 , p1 ) = ∅: and if r7 ∈ γ(r6 ; r4 , r5 ), we have (v2 v3 p2 )3 , (r4 p1 r5 r6 r7 )5 and a 4hole from the remaining points; and if r7 ∈ γ(r6 ; r4 , p2 ), we have ( p1 v2 r5 )3 , (r4 r7 r6 p2 α(r4 ; p2 , p3 ))5 and a 4hole from the remaining points. Suppose r5 ∈ γ(r6 ; p2 , r4 ). If r7 ∈ γ(r6 ; p2 , r4 ), 329
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we have ( p1 v2 r5 )3 , ( p2 r6 r7 r4 α(r4 ; p2 , p3 ))5 and a 4hole from the remaining points. If r7 ∈ γ(r6 ; r4 , p2 ) ∩ H (r6 ; r4 p2 ), we have ( p1 v2 r6 r5 )3 , ( p3 r4 r7 p2 α( p3 ; p2 , v1 ))5 and a 3hole from the remaining points. If r7 ∈ γ( p2 ; r4 , v1 ), we have (v1 v2 p1 )3 , (r1 r6 r5 p2 r7 )5 and a 4hole from the remaining points. If r7 ∈ γ( p2 ; v1 , p3 ), we have ( p1 v2 r5 )3 , (v1 r4 r6 p2 r7 )5 and a 4hole from the remaining points. Secondly, consider r4 ∈ γ(v1 ; p2 , p3 ), then we will discuss the region γ(r4 ; p2 , p3 ) ∩ ( p1 p2 p3 ) = ∅, as shown in Figure 15.
Figure 15. Figure of T1  = 1 and T2  = 1 with shaded region nonempty.
(c2)
Assume γ(r4 ; p2 , p3 ) ∩ ( p1 p2 p3 ) = ∅. We have (r2 r3 v3 )3 , (r1 p3 r4 r6 p2 )5 and a 4hole from the remaining points where r1 = α( p3 ; p2 , v3 ), r6 = α(r4 ; p2 , p1 ). Assume γ(r4 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅. We have L5 ( p2 , r4 ). Assume γ(r4 ; p2 , p3 ) ∩ ( p1 p2 p3 ) = ∅ and γ(r4 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅. Then we may assume r6 ∈ γ(r4 ; p2 , p3 ), r7 ∈ γ(r4 ; p1 , p2 ). Suppose r6 ∈ γ(r4 ; v1 , p3 ) ∩ ( p1 p2 p3 ). If γ(r6 ; r4 , p3 ) = ∅, we have ( p1 r5 p2 r7 )4 , (v1 r4 r6 r1 p3 )5 and (r2 r3 v3 )3 where r1 = α(r6 ; p3 , r4 ). If γ(r6 ; r4 , p3 ) = ∅: and if r7 ∈ γ(r4 ; v1 , p2 ) ∩ γ(v1 ; p2 , r4 ), we have L5 ( p2 ; r4 ); and if r7 ∈ γ(r4 ; r5 , p2 ) ∩ γ( p2 ; p1 , v1 ), we have L5 (r4 ; r7 ); and if r7 ∈ γ(r4 ; p1 , r5 ), we have (v1 v2 p1 )3 , (r4 r7 r5 p2 r6 )5 and a 4hole from the remaining points. Suppose r6 ∈ γ(r4 ; v1 , p2 ) ∩ ( p1 p2 p3 ). If r7 ∈ γ(v1 ; p1 , p2 ) ∩ ( p1 p2 p3 ), we have (r5 v2 p1 )3 , (v1 r7 p2 r6 r4 )5 and a 4hole from the remaining points. If r7 ∈ γ(v1 ; p2 , r4 ) ∩ γ(r4 ; p1 , p2 ): and if r7 ∈ γ(r7 ; r4 , p1 ), we have L5 (r4 , r7 ); and if r5 ∈ γ(r7 ; r4 , p2 ), we have (v1 v2 p1 )3 , (r4 r7 r5 p2 r6 )5 and a 4hole from the remaining points.  T2  = 0. Denote r1 , r2 , r3 ∈ T3 , r4 ∈ T2 , r5 , r6 , r7 ∈ ( p1 p2 p3 ). Let r5 = α( p3 ; p1 , p2 ) within ( p1 p2 p3 ). If γ(r5 ; p1 , p3 ) = ∅, we have L5 (r5 ; α(r5 ; p3 , p1 )). Then we assume γ(r5 ; p1 , p3 ) = ∅, and we will discuss the region γ(r5 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅, as hown in Figure 16.
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Figure 16. Figure of T1  = 1 and T2  = 0.
Assume γ(r5 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅, we have (v1 r4 p3 )3 , (r5 p1 v2 p2 α( p2 ; r5 , p3 ))5 and a 4hole from the remaining points.
Assume γ(r5 ; p2 , p3 ) ∩ ( p1 p2 p3 ) = ∅. Let p6 = α(r5 ; p2 , p3 ). Suppose r4 ∈ γ(r5 ; p3 , r6 ). We have (r2 r3 v3 )3 , ( p2 r1 p3 r4 α(r4 ; p2 , p3 ))5 and a 4hole from the remaining points where r1 = α( p2 ; p3 , v2 ). Suppose r4 ∈ γ(r5 ; p1 , r6 ). We have (r2 r3 r4 )3 , ( p2 r1 p3 r5 r6 )5 and a 4hole from the remaining points where r1 = α( p2 ; p3 , v2 ). Assume γ(r5 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅ and γ(r5 ; p2 , p3 ) ∩ ( p1 p2 p3 ) = ∅. Without loss of generality, we suppose r6 ∈ γ(r5 ; p1 , p2 ), r7 ∈ γ(r5 ; p2 , p3 ).
Firstly, we may assume r6 ∈ γ(r5 ; v2 , p2 ). Suppose r4 ∈ γ(r6 ; p7 , p2 ). We have L5 ( p2 , r6 ). Suppose r4 ∈ γ(r5 ; r6 , p1 ) ∩ H (r5 ; r6 p2 ). We have a cutting line L5 (r5 , r6 ). Suppose r4 ∈ γ(r6 ; p6 , p1 ). If r7 ∈ γ(r4 ; p2 , p3 ), we have (v1 v2 p1 )3 , (v4 r5 r6 p2 r7 )5 and a 4hole from the remaining points. If r7 ∈ γ( p2 ; r4 , p5 ), we have (v3 r2 r3 )3 , (v1 p1 r6 r5 )4 and ( p3 r4 r7 p2 r1 )5 where r1 = α( p2 ; p3 , v2 ).
Secondly, we have may assume r6 ∈ γ(r5 ; v2 , p3 ), we have (v1 p1 r4 )3 , (r5 r6 v2 p2 r7 )5 and a 4hole from the remaining points. (d)
 T1  = 0.  T2  = 0. Let r4 , r5 , r6 , r7 ∈ ( p1 p2 p3 ). And r1 = α( p3 ; p2 , v3 ), r4 = α( p3 ; p2 , p1 ), r5 = α( p2 ; p1 , r4 ). If γ(r4 ; p2 , p3 ) = ∅, we have (r2 r3 v3 )3 , ( p2 r1 p3 r4 α(r4 ; p2 , p3 ))5 and a 4hole from the remaining points. Assume r5 ∈ γ(r4 ; p1 , p3 ). If γ(r5 ; p2 , p1 ) = ∅, we have a cutting line L5 (r5 ; α(r5 ; p2 , p1 )). Then we will discuss the region γ(v4 ; p1 , p2 ) ∩ ( p1 p2 p3 ) and γ(r4 ; p1 , p3 ) ∩ γ( p1 ; p5 , r4 ), as shown in Figure 17.
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Figure 17. Figure of T1  = 1 and T2  = 0 with shaded region nonempty.
Assume γ(v4 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅. We have (r7 r5 v2 )3 , (r4 p3 v1 p1 r6 )5 and a 4hole from the remaining points where r6 = α(r4 ; p1 , p3 ).
Assume γ(r4 ; p1 , p3 ) ∩ γ( p1 ; p5 , r4 ) = ∅. Let r6 = α( p1 ; p3 , r4 ). Suppose r7 ∈ γ(r6 ; r4 , p1 ). We have (v1 v2 p1 )3 , ( p2 r4 r7 r6 r5 )5 and a 4hole from the remaining points. Suppose r7 ∈ γ(r6 ; r4 , v1 ) ∩ γ(r4 ; p1 , p2 ). We have (r1 r2 r3 )3 , ( p1 v2 p2 r5 )4 and (r4 p3 v1 r6 r7 )5 . Suppose r7 ∈ γ(r6 ; r5 , v1 ) ∩ γ(r4 ; p1 , p2 ). We have (v3 r2 r3 )3 , (r4 p2 r1 p3 )4 and (r6 v1 p2 r5 r7 )5 . Suppose r7 ∈ γ(r6 ; r5 , p2 ). We have (v1 v2 p2 )3 , (r4 p3 r6 r7 r5 )5 and a 4hole from the remaining points. Suppose r7 ∈ γ(r6 ; p1 , r3 ). We have a cutting line L5 ( p3 , r6 ).
Assume γ(v4 ; p1 , p2 ) ∩ ( p1 p2 p3 ) = ∅ and γ(r4 ; p1 , p3 ) ∩ γ( p1 ; p5 , r4 ) = ∅. Without loss of generality, assume r6 ∈ γ(r4 ; p1 , p2 ) ∩ ( p1 p2 p3 ), r7 ∈ γ(r4 ; p1 , p3 ) ∩ γ( p1 ; p5 , r4 ).
Suppose r6 ∈ γ(r5 ; p3 , p1 ). We have a cutting line L5 (r6 , α(r6 ; p1 , p3 ).
Suppose r6 ∈ γ(r5 ; p3 , p1 ) ∩ γ(v1 ; r4 , p3 ). If r7 ∈ γ(r5 ; p3 , p2 ) ∩ γ( p1 ; r5 , r4 ), we have (v2 p2 r5 )3 , (v1 p1 r7 r4 r6 )5 and a 4hole from the remaining points. If r7 ∈ γ(r5 ; p3 , p1 ) ∩ γ(r4 ; p1 , p3 ), we have (v2 p2 p3 )3 , (v1 p1 r5 r7 r6 )5 and a 4hole from the remaining points.
Suppose r6 ∈ γ(r5 ; p3 , p1 ) ∩ γ(v1 ; r1 , p1 ). If r7 ∈ γ(r6 ; r4 , v1 ), we have (v1 r6 r7 r4 p3 )5 , ( p1 v2 p2 r5 )4 and (r1 r2 r3 )3 . If r7 ∈ γ(r6 ; r5 , v1 ), we have (v2 v3 p2 )3 , (v1 p1 r5 r7 r6 )5 and a 4hole from the remaining points. (6)
 T3  = 2.
Let r1 , r2 ∈ T3 and r1 = α( p2 ; p3 , v1 ). Assume r2 ∈ γ(r1 ; p2 , v3 ). We have ( p2 r1 r2 v3 )4 and the remaining 9 points are in H (v3 ; p2 p3 ), as shown in Figure 18.
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Figure 18. Figure of T3  = 2.
By the discussion of Part One, we know our conclusion is right. Assume r2 ∈ γ(r1 ; p3 , v2 ). We have ( p3 r1 r2 v2 )4 . By the discussion of Part One, we know our conclusion is also right. Assume r2 ∈ γ(r1 ; p3 , p2 ). We have a cutting line L5 ( p2 , α( p2 ; p3 , p1 )). (7)
 T3  = 1. Let r1 ∈ T3 , r2 ∈ T1 , r3 ∈ T2 and r4 , r5 , r6 , r7 ∈ ( p1 p2 p3 ). Let r4 = α( p3 ; p2 , p1 ) within ( p1 p2 p3 ). Assume r4 ∈ γ( p3 ; p1 , v1 ), as shown in Figure 19.
Figure 19. Figure of T3  = 1.
If r2 ∈ γ(v1 ; p2 , p3 ), we have a cutting line L5 (r2 , α(r2 ; p2 , p1 )). If r2 ∈ γ(v1 ; p2 , p1 ), we have a cutting line L5 (v1 , α(v1 ; p2 , p1 )). Assume r4 ∈ γ( p3 ; p2 , v1 ). If γ(r4 ; p3 , p2 ) = ∅, we have a cutting line L5 (r4 , α(r4 ; p3 , p2 )). If γ(r4 ; p2 , p3 ) = ∅, we have a cutting line L5 (r4 ; α(r4 ; p2 , p3 )). If γ(r4 ; p3 , p2 ) = ∅ and γ(r4 ; p2 , p3 ) = ∅: and if r1 ∈ γ(r4 ; p2 , v3 ), we have (r4 p3 v3 r1 )4 ; and if r1 ∈ γ(r4 ; p3 , v3 ), we have ( p2 r4 r1 v3 )4 . Then the remaining 9 points are all in H (v3 ; p2 p3 ). By the discussion of Part One, our conclusion is right. (8)
 T3  = 0. 333
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Then  T2  = 0,  T1  = 0 and ri ∈ ( p1 p2 p3 ) for i = 1, ..., 7. Let r1 = α( p1 ; p3 , p2 ). If r1 ∈ γ( p1 ; p3 , v3 ), as shown in Figure 20.
Figure 20. Figure of T1  = 0 and T2  = 0.
We have (v1 p1 r1 p3 )4 and the remaining 9 points are all in H ( p3 ; p1 r1 ). By the discussion of Part One, our conclusion is right. If r1 ∈ γ( p1 ; p3 , v3 ): and if γ(r1 ; p1 , p3 ) = ∅, we have (v1 p3 r1 p1 )4 and the remaining 9 points are all in H (v1 ; p3 r1 ); and if γ(r1 ; p1 , p3 ) = ∅, we have a cutting line L5 (r1 , α(r1 ; p1 , p3 )). 4. Conclusions In this paper, we discuss a classical discrete geometry problem. After detailed proof, conclusion shows that a general planar point set contains a 3hole, a 4hole and a 5hole, with at least 13 points. As 30 ≤ n(6) ≤ 463 [16,21] and n(7) does not exist, the proposed theorem will contribute to the theoretical research to some degree. Discrete geometry is a meaningful tool to study social networks. Therefore, our conclusion could be used to deal with some complex network problems. For example, under the environment of competition social structure, the structural holes which have been studied by many economists, are part of an important research branch of discrete geometry. Author Contributions: Conceptualization, Z.Y.; Funding acquisition, Q.Y.; Methodology, Q.Y. and X.Y. Funding: National Social Science Fund of China (18CGL018). Acknowledgments: National Social Science Fund of China (18CGL018). This fund covers the costs to publish in open access. Conﬂicts of Interest: The author declares no conﬂict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8.
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Hosono, K.; Urabe, M. A minimal planar point set with speciﬁed disjoint empty convex subsets. Lect. Notes Comput. Sci. 2008, 4535, 90–100. Hosono, K.; Urabe, M. On the number of disjoint convex quadrilaterals for a planar point set. Comput. Geom. Theory Appl. 2001, 20, 97–104. [CrossRef] Hosono, K.; Urabe, M. On the minimum size of a point set containing two nonintersecting empty convex polygons. Lect. Notes Comput. Sci. 2005, 3742, 117–122. Wu, L.; Ding, R. Reconﬁrmation of two results on disjoint empty convex polygons. Lect. Notes Comput. Sci. 2007, 4381, 216–220. Bhattacharya, B.; Das, S. On the minimum size of a point set containing a 5hole and a disjoint 4hole. Stud. Sci. Math. Hung. 2011, 48, 445–457. [CrossRef] Bhattacharya, B.B. Sandip Das, On pseudoconvex partitions of a planar point set. Discret. Math. 2013, 313, 2401–2408. [CrossRef] Bhattacharya, B.; Das, S. Disjoint Empty Convex Pentagons in Planar Point Sets. Period. Math. Hung. 2013, 66, 73–86. [CrossRef] Hosono, K.; Urabe, M. Specifed holes with pairwise disjoint interiors in planar point sets. AKCE Int. J. Graph. Comb. 2018. [CrossRef] You, X.; Wei, X. On the minimum size of a point set containing a 5hole and double disjoint 3holes. Math. Notes 2013, 97, 951–960. [CrossRef] You, X.; Wei, X. A note on the upper bound for disjoint convex partitions. Math. Notes 2014, 96, 268–274. [CrossRef] You, X.; Wei, X. A note on the value about a disjoint convex partition problem. Ars Comb. 2014, 115, 459–465. You, X.; Chen, T. A note on the value about a disjoint convex partition problem. Math. Notes, 2018, 104, 135–149. Koshelev, V.A. On Erd˝osSzekeres problem for empty hexagons in the plane. Modelirovanie i Analiz Informatsionnykh Sistem 2009, 16, 22–74. c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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MultiGranulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation Zhanao Xue 1,2, *, Minjie Lv 1,2 , Danjie Han 1,2 and Xianwei Xin 1,2 1 2
*
College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China; [email protected] (M.j.L.); [email protected] (D.j.H.); [email protected] (X.w.X.) Engineering Lab of Henan Province for Intelligence Business & Internet of Things, Henan Normal University, Xinxiang 453007, China Correspondence: [email protected]
Received: 7 August 2018; Accepted: 26 September 2018; Published: 28 September 2018
Abstract: From the perspective of the degrees of classiﬁcation error, we proposed graded rough intuitionistic fuzzy sets as the extension of classic rough intuitionistic fuzzy sets. Firstly, combining dominance relation of graded rough sets with dominance relation in intuitionistic fuzzy ordered information systems, we designed typeI dominance relation and typeII dominance relation. TypeI dominance relation reduces the errors caused by single theory and improves the precision of ordering. TypeII dominance relation decreases the limitation of ordering by single theory. After that, we proposed graded rough intuitionistic fuzzy sets based on typeI dominance relation and typeII dominance relation. Furthermore, from the viewpoint of multigranulation, we further established multigranulation graded rough intuitionistic fuzzy sets models based on typeI dominance relation and typeII dominance relation. Meanwhile, some properties of these models were discussed. Finally, the validity of these models was veriﬁed by an algorithm and some relative examples. Keywords: graded rough sets; rough intuitionistic fuzzy sets; dominance relation; logical conjunction operation; logical disjunction operation; multigranulation
1. Introduction Pawlak proposed a rough set model in 1982, which is a signiﬁcant method in dealing with uncertain, incomplete, and inaccurate information [1]. Its key strategy is to consider the lower and upper approximations based on precise classiﬁcation. As a tool, the classic rough set is based on precise classiﬁcation. It is too restrictive for some problems in the real world. Considering this defect of classic rough sets, Yao proposed the graded rough sets (GRS) model [2]. Then researchers paid more attention to it and relative literatures began to accumulate on its theory and application. GRS can be deﬁned as the lower approximation being Rk ( X ) = { x [ x ] R  − [ x ] R ∩ X  ≤ k, x ∈ U } and the upper approximation being Rk ( X ) = { x [ x ] R ∩ X  > k, x ∈ U }. [ x ] R ∩ X  is the absolute number of the elements of [ x ] R  inside X and should be called internal grade, [ x ] R  − [ x ] R ∩ X  is the absolute number of the elements of [ x ] R  outside X and should be called external grade. Rk ( X ) means union of the elements whose equivalence class’ internalgrade about X is greater than k, Rk ( X ) means union of the elements whose equivalence class’ external grade about X is at most k [3]. In the view of granular computing [4], the classic rough set is a singlegranulation rough set. However, in the real world, we need multiple granularities to analyze and solve problems and Qian et al. proposed multigranulation rough sets solving this issue [5]. Subsequently, multigranulation rough sets were extended in References [6–9]. In addition, in the viewpoint of the degrees of classiﬁcation error, Hu et al. and Wang et al. established a novel model of multigranulation graded covering rough sets [10,11]. Simultaneously, Wu et al. constructed graded multigranulation rough sets [12]. Symmetry 2018, 10, 446; doi:10.3390/sym10100446
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References [13–17] discussed GRS in a multigranulation environment. Moreover, for GRS, it has been studied that the equivalence relation has been extended to the dominance relation [13,14], the limited tolerance relation [17] and so forth [10,11]. In general, all these aforementioned studies have naturally contributed to the development of GRS. Inspired by the research reported in References [5,13–17], intuitionistic fuzzy sets (IFS) are also a theory which describe uncertainty [18]. IFS consisting of a membership function and a nonmembership function are commonly encountered in uncertainty, imprecision, and vagueness [18]. The notion of IFS, proposed by Atanassov, was initially developed in the framework of fuzzy sets [19]. Furthermore, it can describe the “fuzzy concept” of “not this and not that”, that is to say, neutral state or neutral degree, thus it is more precise to portray the ambiguous nature of the objective world. IFS theory is applicable in decisionmaking, logical planning, medical diagnosis, machine learning, and market forecasting, etc. Applications of IFS have attracted people’s attention and achieved fruitful results [20–27]. In recent years, IFS have been a hot research topic in uncertain information systems [6,28–30]. For example, in the development of IFS theory, Ai et al. proposed intuitionistic fuzzy line integrals and gave their concrete values in Reference [31]. Zhang et al. researched the fuzzy logic algebraic system and neutrosophic sets as generalizations of IFS in References [23,26,27]. Furthermore, Guo et al. provided the dominance relation of intuitionistic fuzzy information systems [30]. Both rough sets and IFS not only describe uncertain information but also have strong complementarity in practical problems. As such many researchers have studied the combination of rough sets and IFS, namely, rough intuitionistic fuzzy sets (RIFS) and intuitionistic fuzzy rough sets (IFRS) [32]. For example, Huang et al., Gong et al., Zhang et al., He et al., and Tiwari et al. effectively developed IFRS respectively from uncertainty measures, variable precision rough sets, dominance–based rough sets, intervalvalued IFS, and attribute selection [29,30,33–35]. Additionally, Zhang and Chen, Zhang and Yang, Huang et al. studied dominance relation of IFRS [19–21]. With respect to RIFS, Xue et al. provided a multigranulation covering the RIFS model [9]. The above models did not consider the classiﬁcation of some degrees of error [6–9,29,30,33,36] in dominance relation on GRS and dominance relation in intuitionistic fuzzy ordered information systems [37]. Therefore, in this paper, ﬁrstly, we introduce GRS into RIFS to get graded rough intuitionistic fuzzy sets (GRIFS) solving this problem. Then, considering the need for more precise sequence information in the real world, based on dominance relation of GRS and an intuitionistic fuzzy ordered information system, we respectively perform logical conjunction and disjunction operation to gain typeI dominance relation and typeII dominance relation. After that, we use typeI dominance relation and typeII dominance relation thereby replacing equivalence relation to generalize GRIFS. We design two novel models of GRIFS based on typeI dominance relation and typeII dominance relation. In addition, to accommodate a complex environment, we further extend GRIFS models based on typeI dominance relation and typeII dominance relation, respectively, to multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation. These models present a new path to extract more ﬂexible and accurate information. The rest of this paper is organized as follows. In Section 2, some basic concepts of IFS and GRS, RIFS are brieﬂy reviewed, at the same time, we give the deﬁnition of GRS based on dominance relation. In Section 3, we respectively propose two novel models of GRIFS models based on typeI dominance relation and typeII dominance relation and verify the validity of these two models. In Section 4, the basic concepts of multigranulation RIFS are given. Then, we propose multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation, and provide the concepts of optimistic and pessimistic multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation, respectively. In Section 5, we use an algorithm and example to study and illustrate the multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation, respectively. In Section 6, we conclude the paper and illuminate on future research.
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2. Preliminaries Deﬁnition 1 ([22]). Let U be a nonempty classic universe of discourse. U is denoted by: A = {< x, μ A ( x ), νA ( x ) >  x ∈ U }, A can be viewed as IFS on U, where μ A ( x ) : U → [0, 1] and νA ( x ) : U → [0, 1] . μ A ( x ) and νA ( x ) are denoted as membership and nonmembership degrees of the element x in A, satisfying 0 ≤ μ A ( x ) + νA ( x ) ≤ 1. For ∀ x ∈ U, the hesitancy degree is π A ( x ) = 1 − μ A ( x ) − νA ( x ), noticeably, π A ( x ) : U → [0, 1] . ∀ A, B ∈ IFS(U ), the basic operations of A and B are given as follows:
(1) (2) (3) (4) (5)
A ⊆ B ⇔ μ A ( x ) ≤ μ B ( x ), νA ( x ) ≥ νB ( x ), ∀ x ∈ U, A = B ⇔ μ A ( x ) = μ B ( x ), νA ( x ) = νB ( x ), ∀ x ∈ U, A ∪ B = {< x, max{μ A ( x ), μ B ( x )}, min{νA ( x ), νB ( x )} >  x ∈ U }, A ∩ B = {< x, min{μ A ( x ), μ B ( x )}, max{νA ( x ), νB ( x )} >  x ∈ U }, ∼ A = {< x, νA ( x ), μ A ( x ) >  x ∈ U }.
Deﬁnition 2 ([2]). Let (U, R) be an approximation space, assume k ∈ N, where N is the natural number set. Then GRS can be deﬁned as follows: Rk ( X ) = { x [ x ] R  − [ x ] R ∩ X  ≤ k, x ∈ U }, Rk ( X ) = { x [ x ] R ∩ X  > k, x ∈ U }. Rk ( X ) and Rk ( X ) can be considered as the lower and upper approximations of X with respect to the graded k. Then we call the pair ( Rk ( X ), Rk ( X )) GRS. When k = 0, R0 ( X ) = R( X ), R0 ( X ) = R( X ). However, in general, Rk ( X ) Rk ( X ), Rk ( X ) Rk ( X ). In Reference [4], the positive and negative domains of X are given as follows: POS( X ) = Rk ( X ) ∩ Rk ( X ), NEG ( X ) = ¬( Rk ( X ) ∪ Rk ( X )). Deﬁnition 3 ([36]). If we denote R≥ a = {( xi , x j ) ∈ U × U : f ( xi ) ≥ f ( x j ), ∀ a ∈ A } where A is a subset of the attributes set and f ( x ) is the value of attribute a, then [ x ]≥ a is referred to as the dominance class of dominance ≥ relation R≥ a . Moreover, we denote approximation space based on dominance relations by S = (U, R a ). ≥ Deﬁnition 4. Let (U, R≥ a ) be an information approximation. U/R a is the set of dominance classes induced by ≥ ≥ a dominance relation R a , and [ x ] a is called the dominance class containing x. Assume k ∈ N, where N is the natural number set. GRS based on dominance relation can be deﬁned: ≥ ≥ R≥ k ( X ) = { x [ x ] a  − [ x ] a ∩ X  ≤ k, x ∈ U }, ≥
Rk ( X ) = { x [ x ]≥ a ∩ X  > k, x ∈ U }. ≥
When k = 0, ( R0 ( X ), R0≥ ( X )) will be rough sets based on dominance relation. Example 1. Suppose there are nine patients U = { x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 }, they may suffer from a cold. According to their fever, we get U/R≥ a = {{ x1 , x2 , x4 }, { x3 , x8 }, { x6 , x8 }, { x5 , x7 , x8 , x9 }}, X ⊆ U. Then suppose X = { x1 , x2 , x4 , x7 , x9 }, we can obtain GRS based on dominance relation. The demonstration process is given as follows: Suppose k = 1, then we can get, ≥ ≥ ≥ ≥ ≥ ≥ [ x1 ] ≥ a = [ x2 ] a = [ x4 ] a = { x1 , x2 , x4 }, [ x3 ] a = [ x8 ] a = { x3 , x8 }, [ x6 ] a = [ x8 ] a = { x6 , x8 }, ≥ ≥ ≥ ≥ [ x5 ] a = [ x7 ] a = [ x8 ] a = [ x9 ] a = { x5 , x7 , x8 , x9 }.
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≥
Then, we can calculate R1≥ ( X ), R1 ( X ) and POS( X ), NEG ( X ). ≥
R1≥ ( X ) = { x1 , x2 , x4 }, R1 ( X ) = { x1 , x2 , x4 , x5 , x7 , x8 , x9 }. ≥
POS( X ) = R1≥ ( X ) ∩ R1 ( X ) = { x1 , x2 , x4 } ∩ { x1 , x2 , x4 , x5 , x7 , x8 , x9 } = { x1 , x2 , x4 }, ≥
NEG ( X ) = ¬( R1≥ ( X ) ∪ R1 ( X )) = ¬({ x1 , x2 , x4 } ∪ { x1 , x2 , x4 , x5 , x7 , x8 , x9 }) = { x3 , x6 }.
Through the above analysis, we can see x1 , x2 , and x4 patients suffering from a cold disease and x3 and x6 patients not having a cold disease. ≥ When k = 0, ( R0 ( X ), R0≥ ( X )) will be rough sets based on dominance relation. Deﬁnition 5 ([8,32,35]). Let X be a nonempty set and R be an equivalence relation on X. Let B be IFS in X with the membership function μ B ( x ) and nonmembership function νB ( x ). The lower and upper approximations, respectively, of B are IFS of the quotient set X/R with (1) Membership function deﬁned by μ R( B) ( Xi ) = inf{μ B ( x ) x ∈ Xi }, μ R( B) ( Xi ) = sup{μ B ( x ) x ∈ Xi }. (2) Nonmembership function deﬁned by νR( B) ( Xi ) = sup{νB ( x ) x ∈ Xi }, νR( B) ( Xi ) = inf{νB ( x ) x ∈ Xi }. In this way, we can prove R( B) and R( B) are IFS. For ∀ x ∈ Xi , we can obtain,
μ B ( x ) + νB ( x ) ≤ 1, μ B ( x ) ≤ 1 − νB ( x ),sup{μ B ( x ) x ∈ Xi } ≤ sup{1 − νB ( x ) x ∈ Xi },sup{μ B ( x ) x ∈ Xi } ≤ 1 − inf{νB Hence R( B) is IFS. Similarly, we can prove that R( B) is IFS. The RIFS of R( B) and R( B) are given as ollows: R( B) = {< x, inf μ B (y), sup νB (y) >  x ∈ U }, y∈[ x ]i
y∈[ x ]i
R( B) = {< x, sup μ B (y), inf νB (y) >  x ∈ U }. y∈[ x ]i
y∈[ x ]i
3. GRIFS Model Based on Dominance Relation In this section, we propose a GRIFS model based on dominance relation. Moreover, this model contains a GRIFS model based on typeI dominance relation and GRIFS model based on typeII dominance relation, respectively. Then we employ an example to demonstrate the validity of these two models, and ﬁnish by discussing some basic properties of these two models. ≥
Deﬁnition 6 ([37]). If (U, A, V, f ) is an intuitionistic fuzzy ordered information system, so ( R ) a = {( x, y) ∈ U × U  f a (y) ≥ f a ( x ), ∀ a ∈ A} can be called dominance relation in the intuitionistic fuzzy ordered information system.
[ x ]≥ a
= {y( x, y) ∈ ( R )≥ a , ∀ a ∈ A, y ∈ U } = {yμ a (y) ≥ μ a ( x ), νa (y) ≤ νa ( x ), ∀ a ∈ A, y ∈ U }
≥ [ x ]≥ a is dominance class of x in terms of dominance relation ( R ) a .
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3.1. GRIFS Model Based on TypeI Dominance Relation Deﬁnition 7. Let IS≥ = (U, A, V, f ) be an intuitionistic fuzzy ordered information system and R≥ a be a dominance relation of the attribute set A. Suppose X is the GRS of R≥ a on U, a ∈ A, and IFS B on U about I
≥
attribute a satisﬁes dominance relation ( R ) a . The lower approximation R≥ k ( B ) and the upper approximation I
≥I Rk ( B)
with respect to the graded k are given as follows: When k ≥ 1, we can gain,
R≥ k ( B ) = {< x, I
j
inf
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
(μ B (y) ∧ μB (y)),
y∈( ∧
s =1
≥I Rk ( B)
= {< x,
sup j
y∈( ∧
s =1
≥ (([ x ]≥ a )s ∧[ x ] a ))
μB (y) =
sup j
s =1
(μ B (y) ∨ μB (y)),
j
y∈( ∧
s =1
≥ (([ x ]≥ a )s ∧[ x ] a ))
inf
≥ (([ x ]≥ a )s ∧[ x ] a ))
≥
(νB (y) ∨ νB (y)) >  x ∈ U }, (νB (y) ∧ νB (y)) >  x ∈ U }.
≥
 Rk ( X ) ∩ R≥ ¬( Rk ( X ) ∪ R≥ k ( X ) k ( X )) , νB (y) = . U  U 
Obviously, 0 ≤ μB (y) ≤ 1, 0 ≤ νB (y) ≤ 1, j = 1, 2, · · · , n. When k = 0, μB (y) and νB (y) degenerate to be calculated by the classical rough set. However, under these circumstances, the model is still valid, we call this model RIFS based on typeI dominance relation. ≥ Note that, in GRIFS model based on typeI dominance relation, we let [ x ]≥ a and [ x ] a perform a conjunction j
≥ operation ∧, this is to say ≥I means ∧ (([ x ]≥ a ) s ∧ [ x ] a ). s =1
j
≥ Note that, ∧ (([ x ]≥ a )s ∧ [ x ] a ) in GRIFS model based on typeI dominance relation, if x have j dominance s =1
≥ classes [ x ]≥ a of dominance relation R a on GRS, we perform a conjunction operation ∧ of j dominance classes
≥ [ x ]≥ a and [ x ] a .
According to Deﬁnition 7, the following theorem can be obtained. Theorem 1. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, and B be IFS on U. Then a GRIFS model based on typeI dominance relation has these following properties: I
≥I
(1) R ≥ k ( B ) ⊆ B ⊆ R k ( B ), I
≥I
≥I
≥ (2) A ⊆ B, R≥ k ( A ) ⊆ R k ( B ), R k ( A ) ⊆ R k ( B ), I
≥I
I
≥I
≥I
≥I
≥I
≥I
(3) R k ( A ∩ B ) = R k ( A ) ∩ R k ( B ), R k ( A ∪ B ) = R k ( A ) ∪ R k ( B ). Proof. (1) From Deﬁnition 7, we can get, inf
j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
(μ B (y) ∧ μB (y)) ≤ μ B ( x ) ≤
s =1
sup j
sup
j
(μ B (y) ∨ μB (y)) ⇔ μ
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
(νB (y) ∨ νB (y)) ≥ νB ( x ) ≥
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
s =1 j
inf
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
(νB (y) ∧ νB (y)) ⇔ ν
s =1
s =1
≥I
Hence, R≥ k ( B ) ⊆ B ⊆ R k ( B ). I
(2) Based on Deﬁnition 1 and A ⊆ B, Thus we can get, μ A ( x ) ≤ μ B ( x ), νA ( x ) ≥ νB ( x ). ( y ) = ν ( y ). From Deﬁnition 7, we can get, μA (y) = μB (y), νA B
340
I
R≥ k ( B) I
R≥ k ( B)
(x) ≤ μB (x) ≤ μ
( x ) ≥ νB ( x ) ≥ ν
≥I
Rk ( B) ≥I
Rk ( B)
( x ),
( x ),
Symmetry 2018, 10, 446
Then, in the GRIFS model based on typeI dominance relation, we can get, inf
j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a )) s =1
sup j
y∈( ∧
s =1
≥ (([ x ]≥ a )s ∧[ x ] a ))
(μ A (y) ∧ μA (y)) ≤
s =1
( y )) ≥ ( νA ( y ) ∨ νA
sup j
y∈( ∧
s =1
≥I
inf
j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
≥ (([ x ]≥ a )s ∧[ x ] a ))
(μ B (y) ∧ μB (y)) ⇔ μ
I
R≥ k ( A)
(νB (y) ∨ νB (y)) ⇔ ν
I
Rk≥ ( A)
(x) ≤ μ
(x) ≥ ν
I
Rk≥ ( B) I
R≥ k ( B)
( x ),
( x ).
≥I
Thus we can get, Rk ( A) ⊆ Rk ( B). ≥I
≥I
In the same way, we can get, Rk ( A) ⊆ Rk ( B). (3) From Deﬁnition 7, we can get, μ
I
Rk≥ ( A∩ B)
(x) = =μ
ν
I Rk≥ ( A∩ B)
inf
j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
(x) =
s =1 I
R≥ k ( A)
(x) ∧ μ
(μ A∩ B (y) ∧ μA∩ B (y)) = (
I
R≥ k ( B)
j
( x ),
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
=ν
s =1 I
R≥ k ( A)
(x) ∧ ν
I
R≥ k ( B)
(μ A (y) ∧ μA (y))) ∧ (
s =1
( νA ∩ B ( y ) ∨ νA ∩ B ( y )) = (
sup
inf
j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
sup
j
( y ))) ∧ ( ( νA ( y ) ∨ νA
I
I
≥I
(μ B (y) ∧ μB (y)))
sup
j
(νB (y) ∨ νB (y)))
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
s =1
≥ ≥ Thus we can get, R≥ k ( A ∩ B ) = R k ( A ) ∩ R k ( B ).
s =1
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
( x ),
inf
j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
s =1
I
≥I
≥I
In the same way, we can get Rk ( A ∪ B) = Rk ( A) ∪ Rk ( B).
Example 2. In a city, the court administration needs to recruit 3 staff. Applicants who pass the application, preliminary examination of qualiﬁcations, written examination, interview, qualiﬁcation review, political review, and physical examination can be employed. In order to facilitate the calculation, we simplify the enrollment process to qualiﬁcation review, written test, interview. At present, 12 people have passed the preliminary examination of qualiﬁcations, and 9 of them have passed the written examination (administrative professional ability test and application). U = { x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 } is the domain. We can get U/R≥ a = {{ x1 , x2 , x4 }, { x3 , x8 }, { x7 }, { x4 , x5 , x6 , x9 }} according to the “excellent” and “pass” of the two results. In addition, through the interview of 9 people, the following IFS can be obtained, and we suppose X = { x1 , x4 , x5 , x6 , x9 }, X ⊆ U. B=
[0.9, 0] [0.8, 0.1] [0.65, 0.3] [0.85, 0.1] [0.95, 0.05] [0.7, 0.3] [0.5, 0.2] [0.87, 0.1] [0.75, 0.2] . , , , , , , , , x1 x2 x3 x4 x5 x6 x7 x8 x9
To solve the above problems, we can use the model described in References [38,39], which are rough sets based on dominance relation. First, according to U/R≥ a , we can get, ≥
R ≥ ( X ) = { x4 , x5 , x6 , x9 }, R ( X ) = { x1 , x2 , x4 , x5 , x6 , x9 }, Through rough sets based on dominance relation, we can get some applicants with better written test scores. However, regarding IFS B, we cannot use rough sets based on dominance relation to handle the data. Therefore, we are even less able to get the ﬁnal result with the model. To process the interview data, we need to use another model, described in Reference [40]. Through data processing, we can obtain the dominance classes as follows:
≥ ≥ ≥ [ x1 ] ≥ a = { x1 }, [ x2 ] a = { x2 , x4 , x5 , x8 }, [ x3 ] a = { x3 , x4 , x5 , x6 , x8 , x9 }, [ x4 ] a = { x4 , x5 }, ≥ ≥ ≥ ≥ [ x5 ] a = { x5 }, [ x6 ] a = { x6 , x8 , x9 }, [ x7 ] a = { x7 , x8 , x9 }, [ x8 ] a = { x8 }, [ x9 ] ≥ a = { x9 }.
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From the above analysis, we can get, x5 ≥ x1 ≥ x8 ≥ x4 ≥ x2 ≥ x9 ≥ x6 ≥ x3 ≥ x7 Through dominance relation in the intuitionistic fuzzy ordered information system, we can get some applicants with better interview results, but we still cannot get the ﬁnal results. To get this result, we need to analyze the applicants who have better written test scores and better written test scores. Based on the above conclusions, we can determine that only x5 and x4 applicants meet the requirements. However, the performance of others is not certain. If they only need one or two staff, then this analysis can help us to choose the applicant. However, we need 3 applicants, so we cannot get the result in this way. However, there is a model in Deﬁnition 6 that can help us get the results. The calculation process is as follows: According to Example 1, when k = 1, we can get ≥
R1≥ ( X ) = { x1 , x2 , x4 , x5 , x6 , x7 , x9 }, R1 ( X ) = { x1 , x2 , x4 , x5 , x6 , x7 , x9 }, According to Deﬁnitions 7 and 8, we can then get, μB (y) =
≥
≥
 R1 ( X ) ∩ R1≥ ( X ) ¬( R1 ( X ) ∪ R1≥ ( X )) 7 2 = ≈ 0.78, νB (y) = = ≈ 0.22. U  9 U  9
So, according to Deﬁnition 6 and Example 1, we can compute the conjunction operation of [ x ]≥ a
and [ x ]≥ a , and the results are as Table 1.
≥ Table 1. The conjunction operation of [ x ]≥ a and [ x ] a .
x1 x2 x3 x4 x5 x6 x7 x8 x9
[x]a≥
[x]a≥
{ x1 , x2 , x4 } { x1 , x2 , x4 } { x3 , x8 } { x1 , x2 , x4 }, { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x7 } { x3 , x8 } { x4 , x5 , x6 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
x
[x]a≥ ∧[x]a≥
{ x1 } { x2 , x4 } { x3 , x8 } { x4 } { x5 } { x6 , x9 } { x7 } { x8 } { x9 }
GRIFS model based on typeI dominance relation can be obtained as follows: R1≥ ( B) = I
≥I
R1 ( B ) =
#
[0.78,0.22] [0.78,0.22] [0.65,0.3] [0.78,0.22] [0.78,0.22] [0.7,0.3] [0.5,0.22] [0.78,0.22] [0.75,0.22] , , x3 , , , x6 , x7 , , x2 x4 x5 x8 # x1 $x9 [0.9,0] [0.85,0.1] [0.78,0.1] [0.85,0.1] [0.95,0.05] [0.87,0.1] [0.78,0.1] [0.87,0.1] [0.78,0.2] , , , , , , , , . x1 x2 x3 x4 x5 x6 x7 x8 x9
$ ,
≥I
Comprehensive analysis R1≥ ( B) and R1 ( B), we can conclude that x5 , x1 , x8 , x2 and x4 applicants are more suitable for the position in the pessimistic situation. From this example we can see that our model is able to handle more complicated situations than the previous theories, and it can help us get more accurate results. I
3.2. GRIFS Model Based on TypeII Dominance Relation Deﬁnition 8. Let U be a nonempty set and A be the attribute set on U, and a ∈ A, R≥ a is a dominance relation ≥ of attribute A. Let X be GRS of R≥ a on U, and IFS B on U about attribute a satisﬁes dominance relation ( R ) a . The lower and upper approximations of B with respect to the graded k are given as follows: 342
Symmetry 2018, 10, 446
When k ≥ 1, we can get, Π
R≥ k ( B ) = {< x,
inf
j
y∈( ∨
s =1
≥Π
Rk ( B) = {< x,
≥ (([ x ]≥ a )s ∨[ x ] a ))
y∈( ∨
s =1
y∈( ∨
s =1
≥ (([ x ]≥ a )s ∨[ x ] a ))
μB (y) =
(νB (y) ∨ νB (y)) >  x ∈ U },
sup j
(μ B (y) ∨ μB (y)),
sup j
(μ B (y) ∧ μB (y)),
≥ (([ x ]≥ a )s ∨[ x ] a ))
inf
j
y∈( ∨
s =1
≥ (([ x ]≥ a )s ∨[ x ] a ))
≥  Rk ( X ) ∩
(νB (y) ∧ νB (y)) >  x ∈ U }.
≥
R≥ ¬( Rk ( X ) ∪ R≥ k ( X ) k ( X )) , νB (y) = . U  U 
Obviously, 0 ≤ μB (y) ≤ 1, 0 ≤ νB (y) ≤ 1, j = 1, 2, · · · , n. When k = 0, μB (y) and νB (y) are calculated from the classical rough set. However, under these circumstances the model is still valid and we call this model RIFS based on typeII dominance relation. Note that in the GRIFS model based on typeII dominance relation, we perform a disjunction operation ∨ j
≥ ≥ ≥ Π on [ x ]≥ a and [ x ] a , this is to say ≥ means ∨ (([ x ] a )s ∨ [ x ] a ). s =1
j
≥ Note that, ∨ (([ x ]≥ a )s ∨ [ x ] a ) in the GRIFS model based on typeII dominance relation. If x have j s =1
≥ dominance classes [ x ]≥ a of dominance relation R a on GRS, we perform a disjunction operation ∨ of j dominance
≥ classes [ x ]≥ a and [ x ] a , respectively.
According to Deﬁnition 8, the following theorem can be obtained. Π
Theorem 2. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, and B be IFS on U. Then GRIFS model based on typeII dominance relation will have the following properties: ≥Π
Π
(1) R ≥ k ( B ) ⊆ B ⊆ R k ( B ), Π
Π
≥Π
≥Π
≥ (2) A ⊆ B, R≥ k ( A ) ⊆ R k ( B ), R k ( A ) ⊆ R k ( B ), Π
Π
≥Π
Π
≥Π
≥Π
≥ ≥ (3) R ≥ k ( A ∩ B ) = R k ( A ) ∩ R k ( B ), R k ( A ∪ B ) = R k ( A ) ∪ R k ( B ).
Proof. The proving process of Theorem 2 is similar to Theorem 1. Example 3. Nine senior university students are going to graduate from a computer department and they want to work for a famous internet company. Let U = { x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 } be the domain. The company has a campus recruitment at this university. Based on their conﬁdence in programming skills, we get the following IFS B whether they succeed in the campus recruitment or not. At the same time, according to programming skills grades in school, U/R≥ a = {{ x1 , x2 , x4 }, { x4 , x5 , x6 , x9 }, { x3 , x8 }, { x7 }} can be obtained. We suppose X = { x1 , x4 , x5 , x6 , x9 }, X ⊆ U. B=
[0.9, 0] [0.8, 0.1] [0.65, 0.3] [0.85, 0.1] [0.95, 0.05] [0.7, 0.3] [0.5, 0.2] [0.87, 0.1] [0.75, 0.2] , , , , , , , , . x1 x2 x3 x4 x5 x6 x7 x8 x9
We can try to use rough sets based on dominance relation to solve the above problems, as described in Reference [38]. First, according to U/R≥ a , we can get the result as follows. ≥
R ≥ ( X ) = { x4 , x5 , x6 , x9 }, R ( X ) = { x1 , x2 , x4 , x5 , x6 , x9 },
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From the upper and lower approximations, we can get that x4 , x5 , x6 and x9 students may pass the campus interview. However, we cannot use the rough set based on dominance relation to deal with the data of the test scores of their programming skills. In order to process B, we need to use another model, outlined in Reference [40]. The result is as follows: x5 ≥ x1 ≥ x8 ≥ x4 ≥ x2 ≥ x9 ≥ x6 ≥ x3 ≥ x7 Through IFS, we can get that x4 , x2 , x1 and x7 students are better than other students. From the above analysis, we can get student x4 who can be successful in the interview. However, we are not sure about other students. At the same time, from the process of analysis, we ﬁnd that different models are built for the examples, and the predicted results will have deviation. Our model is based on GRS based on dominance relation and the dominance relation in intuitionistic fuzzy ordered information system. Thus, we can use the model to predict the campus interview. Consequently, according to Deﬁnition 8 and Example 1, we can compute the disjunction operation ≥ of [ x ]≥ a and [ x ] a , the results are as Table 2.
≥ Table 2. The disjunction operation of [ x ]≥ a and [ x ] a .
x x1 x2 x3 x4 x5 x6 x7 x8 x9
[x]a≥
[x]a≥
{ x1 , x2 , x4 } { x1 , x2 , x4 } { x3 , x8 } { x1 , x2 , x4 }, { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x7 } { x3 , x8 } { x4 , x5 , x6 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
[x]a≥ ∨[x]a≥
{ x1 , x2 , x4 } { x1 , x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x1 , x2 , x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x8 , x9 } { x7 , x8 , x9 } { x3 , x8 } { x4 , x5 , x6 , x9 }
GRIFS model based on typeII dominance relation can be obtained as follows: Π
R1≥ ( B) = ≥Π
R1 ( B ) =
#
$
[0.78,0.22] [0.78,0.22] [0.65,0.3] [0.7,0.22] [0.7,0.22] [0.7,0.3] [0.5,0.22] [0.65,0.3] [0.7,0.3] , , x3 , x , x5 , x6 , x7 , x8 , x9 , x2 4 # x1 $ [0.9,0] [0.95,0] [0.95,0.05] [0.95,0] [0.95,0.05] [0.95,0.05] [0.87,0.1] [0.87,0.1] [0.95,0.05] , x , , , x7 , x8 , . x1 , x2 , x3 x5 x6 x9 4
Through the above analysis, the students’ interviews prediction can be obtained. x4 , x2 and x1 students are better than others. From this example, the model can help us to analyze the same situation though two kinds of dominance relations. Therefore, this example can be analyzed more comprehensively 4. MultiGranulation GRIFS Models Based on Dominance Relation In this section, we give the multigranulation RIFS conception, and then propose optimistic and pessimistic multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation, respectively. These four models are constructed by multiple granularities GRIFS models based on typeI and typeII dominance relation. Finally, we discuss some properties of these models.
344
Symmetry 2018, 10, 446
Deﬁnition 9 ([39]). Let IS =< U, A, V, f > be an information system, A1 , A2 , · · · , Am ⊆ A, and R Ai is an equivalence relation of x in terms of attribute set A. [ x ] Ai is the equivalence class of R Ai , ∀ B ⊆ U, B is IFS. Then the optimistic multigranulation lower and upper approximations of Ai can be deﬁned as follows: m
∑ RO Ai ( B ) = {< x, μ m
∑ RO A ( B)
i =1 m
∑ RO Ai ( B ) = {< x, μ m
∑ RO A ( B)
i =1
μm
∑
i =1
m
RO Ai ( B )
∑ RO A ( B)
( x ), ν m
∑ RO A ( B)
i
i =1
i =1
∑ RO A ( B)
m
∑ RO A ( B)
i
i =1
i =1y∈[ x ]
i =1
i =1y∈[ x ] m
∑ RO A ( B)
Ai
( x ) = ∧ sup νB (y),
i
( x ) = ∧ sup μ B (y), ν m
i
( x ) >  x ∈ U },
i
( x ) = ∨ inf μ B (y), ν m i =1y∈[ x ] A
( x ) >  x ∈ U },
i
i =1
m
μm
i =1
( x ), ν m
i
i =1
Ai
( x ) = ∨ inf νB (y). i =1y∈[ x ] A
i
i
where [ x ] Ai is the equivalence class of x in terms of the equivalence relation Ai . [ x ] A1 , [ x ] A2 , · · · , [ x ] Am are m equivalence classes, and ∨ is a disjunction operation. Deﬁnition 10 ([39]). Let IS =< U, A, V, f > be an information system, A1 , A2 , · · · , Am ⊆ A, and R Ai is an equivalence relation of x in terms of attribute set A. [ x ] Ai is the equivalence class of R Ai , ∀ B ⊆ U, B is IFS. Then the pessimistic multigranulation lower and upper approximations of Ai can be easily obtained by: m
p
∑ R Ai ( B) = {< x, μ m m
p
p
∑ R A ( B)
( x ), ν m
p
∑ R A ( B)
i =1
m
∑ R PA ( B) i
∑ R PA ( B)
i
i =1
m
μm
∑ R PA ( B)
i =1y∈[ x ]
i =1
i =1y∈[ x ] m
∑ R PA ( B)
Ai
( x ) = ∨ sup νB (y),
i
( x ) = ∨ sup μ B (y), ν m
i
( x ) >  x ∈ U },
i
m
( x ) = ∧ inf μ B (y), ν m i =1y∈[ x ] A
( x ) >  x ∈ U },
i
i =1
i
i =1
μm
i =1
p
∑ R A ( B)
i =1
( x ), ν m
i
i =1
∑ R Ai ( B) = {< x, μ m
i =1
p
∑ R A ( B)
i =1
Ai
( x ) = ∧ inf νB (y). i =1y∈[ x ] A
i
i
where [ x ] Ai is the equivalence class of x in terms of the equivalence relation Ai . [ x ] A1 , [ x ] A2 , · · · , [ x ] Am are m equivalence classes, and ∧ is a conjunction operation. 4.1. GRIFS Model Based on TypeI Dominance Relation Deﬁnition 11. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, A1 , A2 , · · · , ≥ Am ⊆ A. ( R≥ a )i is a dominance relation of x in terms of attribute Ai , a ∈ Ai , where ([ x ] a )i is the dominance ≥ ≥ class of ( R a )i . Suppose X is GRS of ( R a )i and B is IFS on U. IFS B with respect to attribute a satisﬁes ≥ dominance relation (( R ) a )i . Therefore, the lower and upper approximations of B with respect to the graded k are given as follows: When k ≥ 1, we can get, I
m
≥I
i =1
(k)
m
≥I
i =1
(k)
∑ RO Ai
∑ RO Ai
μBi (y) =
( B) = {< x, μ m
∑ RO A
i =1
( B) = {< x, μ
i
i =1
 Rk ( X )∩ R≥ k ( X ) , U 
i
( B)
( x ), ν m
∑ RO A
i =1
(k) ≥I
m
∑ RO A
≥
≥I
(k)
( x ), ν ( B)
νB i (y) = 345
i
i =1
i
( B)
( x ) >  x ∈ U },
(k) ≥I
m
∑ RO A
≥
≥I
(k)
( x ) >  x ∈ U }, ( B)
¬( Rk ( X )∪ R≥ k ( X )) . U 
Symmetry 2018, 10, 446
We can get GRS in A1 , A2 , · · · , Am , then there will be μB1 (y), μB2 (y), μB3 (y), · · · , μBm (y) and νB 1 (y), νB 2 (y), νB 3 (y), · · · , νB m (y). Subsequently, we can obtain, m
μm
∑ RO A
i =1
μ
≥I i
( B)
(x) = ∨
i =1
inf
j
s =1
(k)
m
≥I
m
∑ RO A
i =1
i
(k)
(x) = ∧
i =1
( B)
∑ RO A
(μ B (y) ∨ μBi (y)),
j
s =1
i =1
i
sup ≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
m
(μ B (y) ∧ μBi (y)), ν m
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
ν
i
( B)
(x) = ∧
i =1
(k)
m
≥I
m
∑ RO A
i =1
i
≥I
i
(k)
(x) = ∨
i =1
( B)
(νB (y) ∨ νB i (y)),
sup j
y∈( ∧
s =1 j
≥ (([ x ]≥ a )s ∧[ x ] a ))
inf
i
(νB (y) ∧ νB i (y)).
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a )) s =1
i
Obviously, 0 ≤ μB (y) ≤ 1, 0 ≤ νB (y) ≤ 1, j = 1, 2, · · · , n. m
≥I
i =1
(k)
When ∑ RO Ai
m
≥I
i =1
(k)
( B ) = ∑ RO Ai
( B), B is an optimistic multigranulation GRIFS model based on typeI
dominance relation. When k = 0, μBi (y) and νB i (y) are calculated through the classical rough set. However, under these circumstances the model is still valid and we call this model an optimistic multigranulation RIFS based on typeI dominance relation. Deﬁnition 12. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, A1 , A2 , · · · , ≥ Am ⊆ A. ( R≥ a )i is a dominance relation of x in terms of attribute Ai , where ([ x ] a )i is the dominance class of ≥ ≥ ( R a )i . Suppose X is GRS of ( R a )i and B is IFS on U. IFS B about attribute a satisﬁes dominance relation (( R )≥ a )i , a ∈ Ai . Then the lower and upper approximations of B with respect to the graded k are given as follows: When k ≥ 1, we can get, I
m
p
∑ R Ai
i =1 m
∑
i =1
p RA i
≥I (k)
( B) = {< x, μ m
≥I
p
∑ RA
i =1
i
≥I (k)
( B) = {< x, μ
μBi (y) =
m
i =1
i
( x ), ν m
(k)
( x ), ν ( B)
m
≥I
p
∑ RA
i =1
(k) ≥I
p
∑ RA
( B)
i
i =1
≥
i
( x ) >  x ∈ U },
(k) ≥I
p
∑ RA
( B)
(k)
( x ) >  x ∈ U }, ( B)
≥
 Rk ( X ) ∩ R≥ ¬( Rk ( X ) ∪ R≥ k ( X ) k ( X )) , νB i (y) = . U  U 
We can obtain GRS in A1 , A2 , · · · , Am , then there will be μB1 (y), μB2 (y), μB3 (y), · · · , μBm (y) and , νB m (y). Subsequently, we can obtain,
νB 1 (y), νB 2 (y), νB 3 (y), · · · μm
m
i =1
μ
≥I
p
∑ RA
i
( B)
(x) = ∧
i =1
j
y∈( ∧
s =1
(k)
≥ (([ x ]≥ a )s ∧[ x ] a ))
m
m
i =1
≥I
p
∑ RA
i
(k)
(x) = ∨
i =1
( B)
Obviously, 0 ≤ p
When ∑ R A i i =1
≥I (k)
μB (y)
(μ B (y) ∨ μBi (y)), ν
i =1
p
≥I (k)
νB (y)
i
( B)
(x) = ∨
i =1
(k)
m
≥I
p
∑ RA
i
(k)
s =1
(x) = ∧ ( B)
i =1
(νB (y) ∨ νB i (y)),
sup j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a ))
m
i =1
i
≤ 1, 0 ≤
( B ) = ∑ R Ai
i =1
i
m
m
≥I
p
∑ RA
sup j
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a )) s =1
m
(μ B (y) ∧ μBi (y)), ν m
inf
j
inf
(νB (y) ∧ νB i (y)).
≥ y∈( ∧ (([ x ]≥ a )s ∧[ x ] a )) s =1
i
i
≤ 1, j = 1, 2, · · · , n.
( B), B is a pessimistic multigranulation GRIFS model based on typeI
dominance relation. When k = 0, μBi (y) and νB i (y) are calculated through the classical rough set. However, under these circumstances the model is still valid and we call this model a pessimistic multigranulation RIFS based on typeI dominance relation.
346
Symmetry 2018, 10, 446
j
≥ Note that, ( ∧ (([ x ]≥ a )s ∧ [ x ] a )) in multigranulation GRIFS models based on typeI dominance relation. s =1
i
≥ If x have j dominance classes [ x ]≥ a of dominance relation R a on GRS, we perform a conjunction operation ∧ of j
≥ dominance classes [ x ]≥ a and [ x ] a , respectively. Note that multigranulation GRIFS models based on typeI dominance relation are formed by combining multiple granularities GRIFS models based on typeI dominance relation.
According to Deﬁnitions 11 and 12, the following theorem can be obtained. Theorem 3. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, A1 , A2 , · · · , Am ⊆ A, and B be IFS on U. Then the optimistic and pessimistic multigranulation GRIFS models based on typeI dominance relation have the following properties: I
m
∑ ROAi
i =1
m
∑ RA
i
m
( B ) = ∪ R Ai ≥ ( B ), I
(k)
i =1
(k) ≥I
p
i =1
≥I
m
( B ) = ∩ R Ai ≥ ( B ), I
(k)
i =1
(k)
≥I
m
∑ ROA
i =1
i
m
(k)
i =1
≥I
m
∑ RA
m
p
i =1
i
(k)
≥I
( B ) = ∩ R Ai ( k ) ( B ).
≥I
( B ) = ∪ R Ai ( k ) ( B ). i =1
Proof. One can derive them from Deﬁnitions 7, 11, and 12. 4.2. GRIFS Model Based on TypeII Dominance Relation Π
Deﬁnition 13. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, A1 , A2 , · · · , Am ⊆ A, and U be the universe of discourse. ( R≥ a )i is a dominance relation of x in terms of attribute ≥ ≥ Ai , a ∈ Ai , where ([ x ]≥ ) is the dominance class of ( R a )i . Suppose X is GRS of ( R a )i and B is IFS on U. a i ≥ IFS B about attribute a satisﬁes dominance relation (( R ) a )i . So the lower and upper approximations of B with respect to the graded k are given as follows: When k ≥ 1, we can get, m
≥Π
i =1
(k)
m
≥Π
i =1
(k)
∑ RO Ai
∑ RO Ai
μBi (y)
=
( B) = {< x, μ m
∑ RO A
i =1
( B) = {< x, μ ≥
i =1
i
( B)
( x ), ν m
∑ RO A
i =1
(k) ≥Π
m
∑ RO A
 Rk ( X )∩ R≥ k ( X ) , U 
≥Π i
(k)
νB i (y)
( x ), ν ( B)
=
i =1
i
( B)
( x ) >  x ∈ U },
(k) ≥Π
m
∑ RO A
≥
≥Π i
(k)
( x ) >  x ∈ U }, ( B)
¬( Rk ( X )∪ R≥ k ( X )) . U 
We can obtain GRS in A1 , A2 , · · · , Am , then there will be μB1 (y), μB2 (y), μB3 (y), · · · , μBm (y) and , νB m (y). Subsequently, we can obtain,
νB 1 (y), νB 2 (y), νB 3 (y), · · · m
μm
∑ RO A
i =1
μ
≥Π i
( B)
(x) = ∨
i =1
j
y∈( ∨
s =1
(k)
m
≥Π
m
∑ RO A
i =1
i
(k)
(x) = ∧ ( B)
i =1
≥ (([ x ]≥ a )s ∨[ x ] a ))
∑ RO A
y∈( ∨
s =1
≥ (([ x ]≥ a )s ∨[ x ] a ))
i =1
i
(μ B (y) ∨ μBi (y)), ν
sup j
m
(μ B (y) ∧ μBi (y)), ν m
inf
i
( B)
i =1
(k) ≥Π
m
∑ RO A
i
(k)
(x) = ∨ ( B)
Obviously, 0 ≤ μB (y) ≤ 1, 0 ≤ νB (y) ≤ 1, j = 1, 2, · · · , n.
347
(x) = ∧ m
i =1
i
≥Π
i =1
(νB (y) ∨ νB i (y)),
sup
j
≥ y∈( ∨ (([ x ]≥ a )s ∨[ x ] a )) s =1 j
inf
(νB (y) ∧ νB i (y)).
≥ y∈( ∨ (([ x ]≥ a )s ∨[ x ] a )) s =1
i
i
Symmetry 2018, 10, 446
m
≥Π
i =1
(k)
When ∑ RO Ai
m
≥Π
i =1
(k)
( B ) = ∑ RO Ai
( B), B is an optimistic multigranulation GRIFS model based on typeII
dominance relation. When k = 0, μBi (y) and νB i (y) are calculated from the classical rough set. Under these circumstances, the model is still valid. Π
Deﬁnition 14. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, A1 , A2 , · · · , ≥ Am ⊆ A. ( R≥ a )i is a dominance relation of x in terms of attribute Ai , a ∈ Ai , where ([ x ] a )i is the dominance ≥ class of ( R≥ ) . Suppose X is GRS of ( R ) on U and B is IFS on U. IFS B with respect to attribute a satisﬁes a i a i ≥ dominance relation (( R ) a )i . Then lower and upper approximations of B with respect to the graded k are as follows: When k ≥ 1, we can get, m
∑ RA
≥Π
p
i =1
i
( B) = {< x, μ m
≥Π
p
∑ RA
(k)
i =1
i
( B)
( x ), ν m
i =1
(k)
≥Π
p
∑ RA
i
( B)
( x ) >  x ∈ U },
(k)
≥Π
m
∑ R Ai p
i =1
( B) = {< x, μ (k)
m
i =1
μBi (y) =
≥  Rk ( X ) ∩
≥Π
p
∑ RA
i
(k)
R≥ k ( X )
U 
( x ), ν ( B)
m
i =1
, νB i (y) =
≥Π
p
∑ RA
i
(k)
( x ) >  x ∈ U }, ( B)
≥
¬( Rk ( X ) ∪ R≥ k ( X )) . U 
We can obtain GRS in A1 , A2 , · · · , Am , then there will be μB1 (y), μB2 (y), μB3 (y), · · · , μBm (y) and νB 1 (y), νB 2 (y), νB 3 (y), · · · , νB m (y). Subsequently, we can obtain, μm
∑
i =1
μ
m
p RA i
≥Π
( B)
(x) = ∧
i =1
j
y∈( ∨
s =1
(k)
m
m
p
∑ RA
i =1
≥Π i
(k)
(x) = ∨
i =1
( B)
(μ B (y) ∧ μBi (y)), ν m
inf
≥ (([ x ]≥ a )s ∨[ x ] a ))
∑
(μ B (y) ∨ μBi (y)), ν
j
≥ y∈( ∨ (([ x ]≥ a )s ∨[ x ] a )) s =1
i =1
i
sup
≥Π
( B)
(x) = ∨
i =1
(k)
m
m
≥Π
p
∑ RA
i =1
i
m
p RA i
i
(k)
(x) = ∧ ( B)
i =1
(νB (y) ∨ νB i (y)),
sup j
y∈( ∨
s =1 j
≥ (([ x ]≥ a )s ∨[ x ] a ))
inf
(νB (y) ∧ νB i (y)).
≥ y∈( ∨ (([ x ]≥ a )s ∨[ x ] a )) s =1
i
i
Obviously, 0 ≤ μB (y) ≤ 1, 0 ≤ νB (y) ≤ 1, j = 1, 2, · · · , n. m
When ∑
i =1
p RA i
≥Π
m
( B) = ∑
i =1
(k)
p RA i
≥Π (k)
( B), B is a pessimistic multigranulation GRIFS model based on typeII
dominance relation. When k = 0, μBi (y) and νB i (y) are calculated from the classical rough set. Under these circumstances, the model is still valid. j
≥ ≥ ≥ Note that, in ( ∨ (([ x ]≥ a )s ∨ [ x ] a )) , if x have j dominance classes [ x ] a of dominance relation R a on s =1
i
≥ GRS, we perform a disjunction operation ∨ of j dominance classes [ x ]≥ a and [ x ] a , respectively. Note that multigranulation GRIFS models based on typeII dominance relation are formed by combining multiple granularities GRIFS models based on typeII dominance relation.
According to Deﬁnitions 13 and 14, the following theorem can be obtained.
348
Symmetry 2018, 10, 446
Π
Theorem 4. Let IS≥ =< U, A, V, f > be an intuitionistic fuzzy ordered information system, A1 , A2 , · · · , Am ⊆ A, and IFS B ⊆ U. Then optimistic and pessimistic multigranulation GRIFS models based on typeII dominance relation have the following properties: ≥Π
m
∑
i =1
m
∑
i =1
RO Ai
( B) =
(k)
p RA i
≥Π
( B) =
(k)
m
Π ∪ R Ai ≥ ( B ), (k) i =1
m
Π ∩ R Ai ≥ ( B ), (k) i =1
≥Π
m
∑
i =1
RO Ai
(k)
i =1
≥Π
m
∑
i =1
p RA i
(k)
Proof. One can derive them from Deﬁnitions 7, 13 and 14.
≥Π
m
( B ) = ∩ R Ai ( k ) ( B ),
≥Π
m
( B ) = ∪ R Ai ( k ) ( B ). i =1
5. Algorithm and Example Analysis 5.1. Algorithm Through Examples 1–3, we can conclude that the GRIFS model is effective, and now we use multigranulation GRIFS models based on dominance relation to predict results under the same situations again as Algorithm 1. Algorithm 1. Computing multigranulation GRIFS models based on dominance relation. Input: IS =< U, A, V, f >, X ⊆ U, IFS B ⊆ U, k is a natural number Output: Multigranulation GRIFS models based on dominance relation 1:if (U = φ and A = φ) 2: if can build up GRS 3: if (k ≥ 1 && i = 1 to m && a ∈ Ai ) ≥ 4: compute μB (y) and νB (y), [ x ]≥ a and [ x ] a , for each Ai ⊆ A; 5: then compute ∧ and ∨ of μ B (y) and μ Bi (y), νB i (y) and νB (y) and compute ∧ and ∨ of [ x ]≥ a and
[ x ]≥ a ;
6:
j
s =1
i
for (i = 1 to m && 1 ≤ j ≤ n) compute μ m ( x ), ν m ≥•
7: 8:
∑ RΔA
i =1
9:
i
∑ RΔA
( B)
i =1
(k)
≥• i
( B)
( x ), μ m
∑ RΔA
i =1
(k)
≥• i
(k)
( B)
( x ), ν m
∑ RΔA
i =1
≥• i
(k)
( B)
( x );
end m
≥•
i =1
(k)
compute ∑ RΔAi
10: 11:
≥ if ( x ∈ U && ∀y ∈ ( ∗ (([ x ]≥ a )s ∗ [ x ] a )) )
m
≥•
i =1
(k)
( B), ∑ RΔAi
( B ).
end
12: end 13: end 14:else return NULL 15:end
Note that Δ represents optimistic and pessimistic and ∗ means ∧ or ∨ operation, and in j
j
j
j
≥ ≥ ≥ ≥ ≥ ≥ ≥ ( ∗ (([ x ]≥ a )s ∗ [ x ] a )) , ( ∗ (([ x ] a )s ∗ [ x ] a )) represent ( ∧ (([ x ] a )s ∧ [ x ] a )) or ( ∨ (([ x ] a )s ∨ [ x ] a )) s =1
i
s =1
in this algorithm. • represents I or II.
s =1
i
349
i
s =1
i
Symmetry 2018, 10, 446
Through this algorithm, we next illustrate these models by example again. 5.2. An Illustrative Example We use this example to illustrate Algorithm 1 of multigranulation GRIFS models based on typeI and typeII dominance relation. According to Algorithm 1, we will not discuss this case where k is 0. There are 9 patients. Let U = { x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 } be the domain. Next, we analyzed these 9 patients from these symptoms of fever and salivation. The set of condition attributes are A = {fever, salivation, streaming nose}. For fever, we can get U/R≥ = {{ x1 , x2 , x4 }, { x3 , x8 }, { x7 }, { x4 , x5 , x6 , x9 }}, for salivation there is U/R≥ = {{ x1 }, { x1 , x4 }, { x3 , x5 , x6 }, { x5 , x6 }, { x6 , x9 }, { x2 , x7 , x8 }}, and for streaming nose U/R≥ = {{ x1 }, { x1 , x2 , x4 }, { x3 , x5 , x6 }, { x4 , x6 , x7 , x9 }, { x2 , x7 , x8 }}. According to the cold disease, these patients have the have the following IFS B=
[0.9, 0] [0.8, 0.1] [0.65, 0.3] [0.85, 0.1] [0.95, 0.05] [0.7, 0.3] [0.5, 0.2] [0.87, 0.1] [0.75, 0.2] . , , , , , , , , x1 x2 x3 x4 x5 x6 x7 x8 x9
Suppose X = { x1 , x4 , x5 , x6 , x9 }, k = 1. Then we can obtain multigranulation GRIFS models based on typeI and typeII dominance relation through Deﬁnitions 11–14. Results are as follows. For fever, according to U/R≥ , we can get, ≥
R1≥ ( X ) = { x1 , x2 , x4 , x5 , x6 , x7 , x9 }, R1 ( X ) = { x1 , x2 , x4 , x5 , x6 , x7 , x9 }, μB1 (y) =
≥  R1 ( X )∩ R1≥ ( X )
U 
=
7 9
≈ 0.78, νB 1 (y) =
≥
¬( R1 ( X )∪ R1≥ ( X )) U 
=
2 9
≈ 0.22.
Similarly, for salivation and streaming nose, the results are as follows: μB2 (y) =
6 3 8 1 ≈ 0.67, νB 2 (y) = ≈ 0.33. μB2 (y) = ≈ 0.89, νB 2 (y) = ≈ 0.11. 9 9 9 9
According to Deﬁnitions 11–14, we can obtain multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation. For μB1 (y) and νB 1 (y), μB2 (y) and νB 2 (y) and μB3 (y) and νB 3 (y), the results are the followings as Table 3. Table 3. The conjunction and disjunction operation of μ B (y) and μB1 (y). x
x1
x2
x3
x4
x5
x6
x7
x8
x9
μ B (y) μ B (y) μ B (y) ∧ μB1 (y) νB (y) ∧ νB 1 (y) μ B (y) ∨ μB1 (y) νB (y) ∧ νB 1 (y) μ B (y) ∧ μB2 (y) νB (y) ∨ νB 2 (y) μ B (y) ∨ μB2 (y) νB (y) ∧ νB 2 (y) μ B (y) ∧ μB3 (y) νB (y) ∨ νB 3 (y) μ B (y) ∨ μB3 (y) νB (y) ∧ νB 3 (y)
0.9 0 0.78 0.22 0.9 0 0.67 0.33 0.89 0.11 0.9 0 0.9 0
0.8 0.1 0.78 0.22 0.8 0.1 0.67 0.33 0.8 0.1 0.8 0.11 0.89 0.1
0.65 0.3 0.65 0.3 0.78 0.22 0.65 0.33 0.67 0.3 0.65 0.3 0.89 0.11
0.85 0.1 0.78 0.22 0.85 0.1 0.67 0.33 0.85 0.1 0.85 0.11 0.89 0.1
0.95 0.05 0.78 0.22 0.95 0.05 0.67 0.33 0.95 0.05 0.89 0.11 0.95 0.05
0.7 0.3 0.7 0.3 0.78 0.22 0.67 0.33 0.7 0.3 0.7 0.3 0.89 0.11
0.5 0.2 0.5 0.22 0.78 0.2 0.5 0.33 0.67 0.2 0.5 0.2 0.89 0.11
0.87 0.1 0.72 0.22 0.78 0.1 0.6 0.33 0.67 0.1 0.87 0.11 0.89 0.1
0.75 0.2 0.75 0.22 0.78 0.2 0.67 0.33 0.75 0.2 0.75 0.2 0.89 0.11
≥ Then, according to Deﬁnition 6, for B, we can get [ x ]≥ a . Then, the conjunction operation of [ x ] a ≥
and [ x ] a can be computed as Table 1.
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Symmetry 2018, 10, 446
For fever, we can get GRIFS based on typeI dominance relation as follows: R1≥ ( B) = I
≥I R1 ( B )
#
[0.78,0.22] [0.78,0.22] [0.65,0.3] [0.78,0.22] [0.78,0.22] [0.7,0.3] [0.5,0.22] [0.78,0.22] [0.75,0.22] , , x3 , , , x6 , x7 , , x2 x4 x5 x8 x9 # x1 $ [0.9,0] [0.85,0.1] [0.78,0.1] [0.85,0.1] [0.95,0.05] [0.87,0.1] [0.78,0.1] [0.87,0.1] [0.78,0.2] . = , , , , , , , , x1 x2 x3 x4 x5 x6 x7 x8 x9
≥ For streaming nose, similar to Table 1, we can obtain [ x ]≥ a ∧ [ x ] a as Table 4.
≥ Table 4. The conjunction operation of [ x ]≥ a and [ x ] a .
x1 x2 x3 x4 x5 x6 x7 x8 x9
[x]a≥
[x]a≥
{ x1 }, { x1 , x2 , x4 } { x1 , x2 , x4 }, { x2 , x7 , x8 } { x3 , x5 , x6 } { x1 , x2 , x4 }, { x4 , x6 , x7 , x9 } { x3 , x5 , x6 } { x3 , x5 , x6 }, { x4 , x6 , x7 , x9 } { x2 , x7 , x8 }, { x4 , x6 , x7 , x9 } { x2 , x7 , x8 } { x4 , x6 , x7 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
x
[x]a≥ ∧[x]a≥
{ x1 } { x2 } { x3 , x5 , x6 } { x4 } { x5 } { x6 } { x7 } { x8 } { x9 }
≥ For salivation, similar to Table 1, we can obtain [ x ]≥ a ∧ [ x ] a as Table 5.
≥ Table 5. The conjunction operation of [ x ]≥ a and [ x ] a .
x
[x]a≥
[x]a≥
x1 x2 x3 x4 x5 x6 x7 x8 x9
{ x1 , x2 } { x1 , x2 }, { x2 , x4 } { x3 , x8 } { x2 , x4 } { x5 , x6 } { x5 , x6 } { x7 , x9 }, { x7 , x8 , x9 } { x3 , x8 }, { x7 , x8 , x9 } { x7 , x9 }, { x7 , x8 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
[x]a≥ ∧[x]a≥
{ x1 } { x2 } { x3 , x8 } { x4 } { x5 } { x6 } { x7 , x9 } { x8 } { x9 }
For streaming nose, we can get GRIFS based on typeI dominance relation as follows: R1≥ ( B) = I
#
[0.9,0.11] [0.8,0.11] [0.65,0.3] [0.85,0.11] [0.89,0.11] [0.7,0.3] [0.5,0.2] [0.87,0.11] , x3 , , , x6 , x7 , , x1 $, x2 x4 x5 x8 [0.75,0.2] , x9
#
≥I
[0.9,0] [0.89,0.1] [0.95,0.05] [0.89,0.1] [0.95,0.05] [0.89,0.11] [0.89,0.11] [0.89,0.1] , , x , , , , x8 , x1 , $ x2 x3 x5 x6 x7 4 [0.89,0.11] . x9
R1 ( B ) =
For salivation, we can get GRIFS based on typeI dominance relation as follows: R1≥ ( B) = I
≥I
#
[0.67,0.33] [0.67,0.33] [0.65,0.33] [0.67,0.33] [0.67,0.33] [0.67,0.33] [0.67,0.33] [0.67,0.33] , , , , , , , x1 $, x2 x3 x4 x5 x6 x7 x8 [0.67,0.33] , x9
R1 ( B ) =
#
[0.9,0] [0.8,0.1] [0.87,0.1] [0.85,0.1] [0.95,0.05] [0.7,0.3] [0.75,0.2] [0.87,0.1] [0.75,0.2] , x , , x6 , x7 , x8 , x1 , x2 , x3 x5 x9 4
351
$ .
$ ,
Symmetry 2018, 10, 446
≥ ≥ For [ x2 ]≥ a = { x1 , x2 }, [ x2 ] a = { x2 , x4 } and [ x2 ] a = { x2 , x4 , x5 , x8 }, based on Deﬁnitions 11–14, we should perform the conjunction operation of them, respectively.
≥ ≥ ≥ ([ x1 ]≥ a ∧ [ x1 ] a ) ∧ ([ x1 ] a ∧ [ x1 ] a ) = ({ x1 } ∧ { x1 , x1 , x4 }) ∧ ({ x1 , x2 } ∧ { x1 , x2 , x4 }) = { x1 } ∧ { x1 , x2 } = { x1 }.
Similarly, for x2 , x4 , x6 and x7 , we can get the results as Tables 4 and 5. Therefore, according to the Deﬁnitions 11 and 12 and the above calculations, we can get multigranulation GRIFS models based on a typeI dominance relation as follows: ≥I
3
∑ RO Ai
i =1 3
∑ RO Ai
i =1
( B) =
1 ≥I
( B) = 1
3
≥I
∑ R PAi
i =1
# #
( B) =
i =1
( B) = 1
#
$ ,
$ .
[0.67,0.33] [0.67,0.33] [0.65,0.33] [0.67,0.33] [0.67,0.33] [0.67,0.33] [0.5,0.33] [0.67,0.33] , , , , , , x7 , , x1 x2 x3 x4 x5 x6 x8 [0.67,0.33] x9
≥I
∑ R PAi
[0.9,0] [0.8,0.1] [0.78,0.1] [0.85,0.1] [0.95,0.05] [0.7,0.3] [0.75,0.2] [0.87,0.1] [0.75,0.2] , x , , x6 , x7 , x8 , x1 , x2 , x3 x5 x9 4
#
1
3
[0.89,0.11] [0.8,0.11] [0.65,0.3] [0.85,0.11] [0.89,0.11] [0.7,0.3] [0.67,0.2] [0.87,0.11] [0.75,0.2] , x2 , x3 , , , x6 , x7 , , x1 x4 x5 x8 x9
$
,
[0.89,0] [0.85,0.1] [0.95,0.05] [0.85,0.1] [0.95,0.05] [0.87,0.05] [0.78,0.1] [0.87,0.1] [0.78,0.1] , , x , , , x7 , x8 , x1 , x2 x3 x5 x6 x9 4
$ .
From the above results, Figures 1 and 2 can be drawn as follows: 1
u(GRIFS typeI)fever
0.9
v(GRIFS typeI) fever
0.8 u(optimisitc multigranulation GRIFS typeI) 0.7 v(optimisitc multigraulation GRIFS typeI) 0.6 u(pessimistic multigranulation GRIFS typeI) 0.5 v(pessimistic multigranulation GRIFS typeI) 0.4 u(GRIFS typeI) salivation 0.3
v(GRIFS typeI) salivation
0.2
u(GRIFS typeI) streaming nose
0.1
0 [ౖ
[
[ౘ
[ౙ
[ౚ
[
[
[
[
v(GRIFS typeI) streaming nose
Figure 1. The lower approximation of GRIFS based on typeI dominance relation, as well as optimistic and pessimistic multigranulation GRIFS based on typeI dominance relation.
For Figure 1, we can obtain, μ(y)OI1 ≥ μ(y)GIn1 Θ μ(y)GI f 1 Θ(y)GIs1 ≥ μ(y) PI1 ,ν(y)GIs1 ≥ ν(y) PI1 ≥ ν(y)OI1 = ν(y)GI f 1 ≥ ν(y)GIn1 ;
352
Symmetry 2018, 10, 446
Note: Θ represents ≤ or ≥; μ(y)GI f 1 and ν(y)GI f 1 represent GRIFS typeI dominance relation (fever); μ(y)GIs1 and ν(y)GIs1 represent GRIFS typeI dominance relation (salivation); μ(y)GIn1 and ν(y)GIn1 represent GRIFS typeI dominance relation (streaming nose); μ(y)OI1 and ν(y)OI1 represent optimistic multigranulation GRIFS typeI dominance relation; μ(y) PI1 and ν(y) PI1 represent pessimistic multigranulation GRIFS typeI dominance relation; From Figure 1, we can get that x1 , x2 , x4 , x5 and x8 patients have the disease, and x7 patients do not have the disease. 1
u(GRIFS typeI)fever
0.9
v(GRIFS typeI) fever
0.8 u(optimisitc multigranulation GRIFS typeI) 0.7 v(optimisitc multigraulation GRIFS typeI) 0.6 u(pessimistic multigranulation GRIFS typeI) 0.5 v(pessimistic multigranulation GRIFS typeI) 0.4 u(GRIFS typeI) salivation 0.3
v(GRIFS typeI) salivation
0.2
u(GRIFS typeI) streaming nose
0.1
0
v(GRIFS typeI) streaming nose ¡ſ
¡ƀ
¡Ɓ
¡Ƃ
¡ƃ
¡Ƅ
¡ƅ
¡Ɔ
¡Ƈ
Figure 2. The upper approximation of GRIFS based on typeI dominance relation, as well as optimistic and pessimistic multigranulation GRIFS based on typeI dominance relation.
Then, from Figure 2, we can obtain, μ(y)OI2 = μ(y)GIn2 ≥ μ(y) PI2 Θ μ(y)GIs2 Θ μ(y)GI f 2 ,ν(y)GIs2 ≥ ν(y)OI2 Θ ν(y) PI2 Θ ν(y)GI f 2 ≥ ν(y)GIn2 ;
Note: Θ represents ≤ or ≥; μ(y)GI f 2 and ν(y)GI f 2 represent GRIFS typeI dominance relation (fever); μ(y)GIs2 and ν(y)GIs2 represent GRIFS typeI dominance relation (salivation); μ(y)GIn2 and ν(y)GIn2 represent GRIFS typeI dominance relation (streaming nose); μ(y)OI2 and ν(y)OI2 represent optimistic multigranulation GRIFS typeI dominance relation; μ(y) PI2 and ν(y) PI2 represent pessimistic multigranulation GRIFS typeI dominance relation; From Figure 2, we can get that x1 , x2 , x3 , x4 , x5 , x6 , x8 , and x9 patients have the disease, and x6 patients do not have the disease. For multigranulation GRIFS models based on typeII dominance relation, the calculations for this model are similar to multigranulation GRIFS models based on typeI dominance relation.
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Symmetry 2018, 10, 446
≥ Firstly, for streaming nose, we can compute the disjunction operation of [ x ]≥ a and [ x ] a , and the results are as Table 6.
≥ Table 6. The disjunction operation of [ x ]≥ a and [ x ] a .
x1 x2 x3 x4 x5 x6 x7 x8 x9
[x]a≥
[x]a≥
{ x1 }, { x1 , x2 , x4 } { x1 , x2 , x4 }, { x2 , x7 , x8 } { x3 , x5 , x6 } { x1 , x2 , x4 }, { x4 , x6 , x7 , x9 } { x3 , x5 , x6 } { x3 , x5 , x6 }, { x4 , x6 , x7 , x9 } { x2 , x7 , x8 }, { x4 , x6 , x7 , x9 } { x2 , x7 , x8 } { x4 , x6 , x7 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
x
[x]a≥ ∨[x]a≥
{ x1 , x2 , x4 } { x1 , x2 , x4 , x5 , x7 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x1 , x2 , x4 , x5 , x6 , x7 , x9 } { x3 , x5 , x6 } { x3 , x4 , x5 , x6 , x7 , x8 , x9 } { x2 , x4 , x6 , x7 , x8 , x9 } { x2 , x7 , x8 } { x4 , x6 , x7 , x9 }
≥ Next, for salivation, we can compute the disjunction operation of [ x ]≥ a and [ x ] a , and the results are as Table 7.
≥ Table 7. The disjunction operation of [ x ]≥ a and [ x ] a .
x
[x]a≥
[x]a≥
x1 x2 x3 x4 x5 x6 x7 x8 x9
{ x1 , x2 } { x1 , x2 }, { x2 , x4 } { x3 , x8 } { x2 , x4 } { x5 , x6 } { x5 , x6 } { x7 , x9 }, { x7 , x8 , x9 } { x3 , x8 }, { x7 , x8 , x9 } { x7 , x9 }, { x7 , x8 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
[x]a≥ ∨[x]a≥
{ x1 , x2 } { x1 , x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x2 , x4 , x5 } { x5 , x6 } { x5 , x6 , x8 , x9 } { x7 , x8 , x9 } { x3 , x7 , x8 , x9 } { x7 , x8 , x9 }
≥ Then, for fever, we compute the disjunction operation of [ x ]≥ a and [ x ] a , and these results are shown as Table 8.
≥ Table 8. The disjunction operation of [ x ]≥ a and [ x ] a .
x1 x2 x3 x4 x5 x6 x7 x8 x9
[x]a≥
[x]a≥
{ x1 , x2 , x4 } { x1 , x2 , x4 } { x3 , x8 } { x1 , x2 , x4 }, { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x7 } { x3 , x8 } { x4 , x5 , x6 , x9 }
{ x1 } { x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x4 , x5 } { x5 } { x6 , x8 , x9 } { x7 , x8 , x9 } { x8 } { x9 }
x
[x]a≥ ∨[x]a≥
{ x1 , x2 , x4 } { x1 , x2 , x4 , x5 , x8 } { x3 , x4 , x5 , x6 , x8 , x9 } { x1 , x2 , x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x9 } { x4 , x5 , x6 , x8 , x9 } { x7 , x8 , x9 } { x3 , x8 } { x4 , x5 , x6 , x9 }
For streaming nose, we can get GRIFS based on typeII dominance relation, # $ Π [0.8,0.11] [0.5,0.2] [0.65,0.3] [0.5,0.3] [0.65,0.3] [0.5,0.3] [0.5,0.3] [0.5,0.2] [0.5,0.3] R1≥ ( B) = , x2 , x3 , x , x5 , x6 , x7 , x8 , x9 , x1 4 # $ ≥Π [0.9,0] [0.95,0] [0.95,0.05] [0.95,0] [0.95,0.05] [0.95,0.05] [0.89,0.1] [0.89,0.1] [0.89,0.1] . R1 ( B ) = , x , , , x7 , x8 , x , x2 , x3 x5 x6 x9 1
4
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Symmetry 2018, 10, 446
For salivation, we can get GRIFS based on typeII dominance relation, Π
R1≥ ( B) = ≥Π
R1 ( B ) =
#
[0.67,0.33] [0.67,0.33] [0.65,0.33] [0.67,0.33] [0.67,0.33] [0.67,0.33] [0.5,0.33] [0.5,0.33] , , , , , x7 , x8 , x1 $ , x2 x3 x4 x5 x6 [0.5,0.33] , # x9 $ [0.9,0] [0.95,0] [0.95,0.05] [0.95,0.05] [0.95,0.05] [0.95,0.05] [0.87,0.1] [0.87,0.1] [0.87,0.1] . , , , , x7 , x8 , x1 , x2 , x3 x4 x5 x6 x9
For fever, GRIFS typeII dominance relation can be calculated as follows: Π
R1≥ ( B) = ≥Π
R1 ( B ) =
#
$
[0.78,0.22] [0.78,0.22] [0.65,0.3] [0.7,0.22] [0.7,0.22] [0.7,0.3] [0.5,0.22] [0.65,0.3] [0.7,0.3] , , x3 , x , x5 , x6 , x7 , x8 , x9 , x2 4 # x1 $ [0.9,0] [0.95,0] [0.95,0.05] [0.95,0] [0.95,0.05] [0.95,0.05] [0.87,0.1] [0.87,0.1] [0.95,0.05] . , x , , , x7 , x8 , x1 , x2 , x3 x5 x6 x9 4
Based on Deﬁnitions 13 and 14, the condition of these patients based on multigranulation GRIFS typeII dominance relation can be obtained as follows: ≥Π
3
∑ RO Ai
i =1
1 ≥Π
3
( B) =
∑ RO Ai
i =1 3
1 ≥Π
∑ R PAi
i =1
1
3
∑ R PAi
i =1
( B) =
#
#
≥Π
( B) = 1
#
( B) =
[0.8,0.11] [0.78,0.2] [0.65,0.3] [0.7,0.22] [0.7,0.3] [0.7,0.3] [0.5,0.22] [0.65,0.2] [0.7,0.3] , x2 , x3 , x , x5 , x6 , x7 , x8 , x9 x1 4
$ ,
[0.9,0] [0.95,0] [0.95,0.05] [0.95,0.05] [0.95,0.05] [0.95,0.05] [0.87,0.1] [0.87,0.1] [0.87,0.1] , , , , x7 , x8 , x1 , x2 , x3 x4 x5 x6 x9
$
[0.67,0.33] [0.5,0.33] [0.65,0.33] [0.5,0.4] [0.65,0.33] [0.5,0.33] [0.5,0.33] [0.5,0.33] [0.5,0.33] , x2 , , x , , x6 , x7 , x8 , x1 x3 x5 x9 4
#
[0.9,0] [0.95,0] [0.95,0.05] [0.95,0] [0.95,0.05] [0.95,0.05] [0.89,0.1] [0.89,0.1] [0.95,0.05] , x , , , x7 , x8 , x1 , x2 , x3 x5 x6 x9 4
. $ ,
$ .
Then, from Figure 3, we can obtain, μ(y)OΠ3 ≥ μ(y)GΠ f 3 Θ μ(y)GΠn3 Θ μ(y)GΠs3 ≥ μ(y) PΠ3 ,ν(y) PΠ3 ≥ ν(y)GΠs3 Θ ν(y)OΠ3 Θ ν(y)GΠ f 3 Θ ν(y)GΠn3 ;
Note: Θ represents ≤ or ≥; μ(y)GΠ f 3 and ν(y)GΠ f 3 represent GRIFS typeII dominance relation (fever); μ(y)GΠs3 and ν(y)GΠs3 represent GRIFS typeII dominance relation (salivation); μ(y)GΠn3 and ν(y)GΠn3 represent GRIFS typeII dominance relation (streaming nose); μ(y)OΠ3 and ν(y)OΠ3 represent optimistic multigranulation GRIFS typeII dominance relation; μ(y) PΠ3 and ν(y) PΠ3 represent pessimistic multigranulation GRIFS typeII dominance relation; From Figure 3, we can see that x1 , x2 , x4 patients have the disease, and x3 , x5 , x6 , x7 , x8 , x9 patients do not have the disease. Then, from Figure 4, we can obtain, μ(y)OΠ4 ≥ μ(y)GΠn4 Θ μ(y)GΠ f 4 Θ μ(y)GΠs4 ≥ μ(y) PΠ4 ,ν(y) PΠ4 Θ ν(y)GΠs4 Θ ν(y)GΠ f 4 Θ ν(y)GΠn4 Θ ν(y)OΠ4 ;
Note: Θ represents ≤ or ≥; μ(y)GΠ f 4 and ν(y)GΠ f 4 represent GRIFS typeII dominance relation (fever); μ(y)GΠs4 and ν(y)GΠs4 represent GRIFS typeII dominance relation (salivation); μ(y)GΠn4 and ν(y)GΠn4 represent GRIFS typeII dominance relation (streaming nose); μ(y)OΠ4 and ν(y)OΠ4 represent optimistic multigranulation GRIFS typeII dominance relation; μ(y) PΠ4 and ν(y) PΠ4 represent pessimistic multigranulation GRIFS typeII dominance relation;
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Symmetry 2018, 10, 446
From Figure 4, we can see that x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 and x9 patients have the disease. 0.9
u(GRIFS typeჟ)fever
0.8
v(GRIFS typeჟ) fever
0.7
u(optimisitc multigranulation GRIFS typeჟ)
0.6
v(optimisitc multigraulation GRIFS typeჟ)
0.5
u(pessimistic multigranulation GRIFS typeჟ)
0.4
v(pessimistic multigranulation GRIFS typeჟ)
0.3
u(GRIFS typeჟ) salivation
0.2
v(GRIFS typeჟ) salivation
0.1
u(GRIFS typeჟ) streaming nose
v(GRIFS typeჟ) streaming nose
0 ¡ſ
¡ƀ
¡Ɓ
¡Ƃ
¡ƃ
¡Ƅ
¡ƅ
¡Ɔ
¡Ƈ
Figure 3. The lower approximation of GRIFS based on typeII dominance relation, as well as optimistic and pessimistic multigranulation GRIFS based on typeII dominance relation. 1
u(GRIFS typeჟ)fever
0.9
v(GRIFS typeჟ) fever
0.8 u(optimisitc multigranulation GRIFS typeჟ)
0.7 v(optimisitc multigraulation GRIFS typeჟ) 0.6 u(pessimistic multigranulation GRIFS typeჟ) 0.5 v(pessimistic multigranulation GRIFS typeჟ) 0.4 u(GRIFS typeჟ) salivation 0.3
v(GRIFS typeჟ) salivation 0.2
u(GRIFS typeჟ) streaming nose
0.1
v(GRIFS typeჟ) streaming nose
0 ¡ſ
¡ƀ
¡Ɓ
¡Ƃ
¡ƃ
¡Ƅ
¡ƅ
¡Ɔ
¡Ƈ
Figure 4. The upper approximation of GRIFS based on typeII dominance relation, as well as optimistic and pessimistic multigranulation GRIFS based on typeII dominance relation.
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From Figures 1 and 2, x1 , x2 , x4 and x8 patients have the disease, x6 and x7 patient do not have the disease. From Figures 3 and 4, x1 , x2 and x4 patients have the disease, x3 , x5 , x6 , x7 , x8 and x9 patients do not have the disease. Furthermore, this example proves the accuracy of Algorithm 1. This example analyzes and discusses multigranulation GRIFS models based on dominance relation. From conjunction and disjunction operations of two kinds of dominance classes perspective, we analyzed GRIFS models based on typeI dominance relation and typeII dominance relation and also optimistic and pessimistic multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation, respectively. Through the analysis of this example, the validity of these multigranulation GRIFS models based on typeI dominance relation and typeII dominance relation models can be obtained. 6. Conclusions These theories of GRS and RIFS are extensions of the classical rough set theory. In this paper, we proposed a series of models on GRIFS based on dominance relation, which were based on the combination of GRS, RIFS, and dominance relations. Moreover, these models of multigranulation GRIFS models based on dominance relation were established on GRIFS models based on dominance relation using multiple dominance relations on the universe. The validity of these models was demonstrated by giving examples. Compared with GRS based on dominance relation, GRIFS models based on dominance relation can be more precise. Compared with GRIFS models based on dominance relation, multigranulation GRIFS models based on dominance relation can be more accurate. It can be demonstrated using the algorithm, and our methods provide a way to combine GRS and RIFS. Our next work is to study the combination of GRS and variable precision rough sets on the basis of our proposed methods. Author Contributions: Z.a.X. and M.j.L. initiated the research and wrote the paper, D.j.H. participated in some of the search work, and X.w.X. supervised the research work and provided helpful suggestions. Funding: This research received no external funding. Acknowledgments: This work is supported by the national natural science foundation of China under Grant Nos. 61772176, 61402153, and the scientiﬁc and technological project of Henan Province of China under Grant Nos. 182102210078, 182102210362, and the Plan for Scientiﬁc Innovation of Henan Province of China under Grant No. 18410051003, and the key scientiﬁc and technological project of Xinxiang City of China under Grant No. CXGG17002. Conﬂicts of Interest: The authors declare no conﬂicts of interest.
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Zhang, Y.Q.; Yang, X.B. An intuitionistic fuzzy dominancebased rough set. In Proceedings of the 7th International Conference on Intelligent Computing, Zhengzhou, China, 11–14 August 2011; Springer: Berlin, Germany, 2011. Huang, B.; Zhuang, Y.L.; Li, H.X.; Wei, D.K. A dominance intuitionistic fuzzyrough set approach and its applications. Appl. Math. Model. 2013, 37, 7128–7141. [CrossRef] Zhang, W.X.; Wu, W.Z.; Liang, J.Y.; Li, D.Y. Theory and Method of Rough Sets; Science Press: Beijing, China, 2001. (In Chinese) Wen, X.J. Uncertainty measurement for intuitionistic fuzzy ordered information system. Master’s Thesis, Shanxi Normal University, Linfen, China, 21 March 2015. (In Chinese) Lezanski, P.; Pilacinska, M. The dominancebased rough set approach to cylindrical plunge grinding process diagnosis. J. Intell. Manuf. 2018, 29, 989–1004. [CrossRef] Huang, B.; Guo, C.X.; Zhang, Y.L.; Li, H.X.; Zhou, X.Z. Intuitionistic fuzzy multigranulation rough sets. Inf. Sci. 2014, 277, 299–320. [CrossRef] Greco, S.; Matarazzo, B.; Slowinski, R. An algorithm for induction decision rules consistent with the dominance principle. In Proceedings of the 2nd International Conference on Rough Sets and Current Trends in Computing, Banff, AB, Canada, 16–19 October 2000; Springer: Berlin, Germany, 2011. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Four Operators of Rough Sets Generalized to Matroids and a Matroidal Method for Attribute Reduction Jingqian Wang 1 and Xiaohong Zhang 2, * 1
College of Electrical & Information Engineering, Shaanxi University of Science & Technology, Xi’an 710021, China; [email protected] 2 School of Arts and Sciences, Shaanxi University of Science & Technology, Xi’an 710021, China * Correspondence: [email protected] or [email protected] Received: 8 August 2018; Accepted: 14 September 2018; Published: 19 September 2018
Abstract: Rough sets provide a useful tool for data preprocessing during data mining. However, many algorithms related to some problems in rough sets, such as attribute reduction, are greedy ones. Matroids propose a good platform for greedy algorithms. Therefore, it is important to study the combination between rough sets and matroids. In this paper, we investigate rough sets and matroids through their operators, and provide a matroidal method for attribute reduction in information systems. Firstly, we generalize four operators of rough sets to four operators of matroids through the interior, closure, exterior and boundary axioms, respectively. Thus, there are four matroids induced by these four operators of rough sets. Then, we ﬁnd that these four matroids are the same one, which implies the relationship about operators between rough sets and matroids. Secondly, a relationship about operations between matroids and rough sets is presented according to the induced matroid. Finally, the girth function of matroids is used to compute attribute reduction in information systems. Keywords: rough set; matroid; operator; attribute reduction
1. Introduction Rough set theory was proposed by Pawlak [1,2] in 1982 as a mathematical tool to deal with various types of data in data mining. There are many practical problems have been solved by it, such as rule extraction [3,4], attribute reduction [5–7], feature selection [8–10] and knowledge discovery [11]. In Pawlak’s rough sets, the relationships of objects are equivalence relations. However, it is well known that this requirement is excessive in practice [12,13]. Hence, Pawlak’s rough sets have been extended by relations [14,15], coverings [16–18] and neighborhoods [6,19]. They have been combined with other theories including topology [20], lattice theory [21,22], graph theory [23,24] and fuzzy set theory [25,26]. However, many optimization issues related to rough sets, including attribute reduction, are NPhard. Therefore, the algorithms to deal with them are often greedy ones [27]. Matroid theory [28–30] is a generalization of graph and linear algebra theories. It has been used in information coding [31] and cryptology [32]. Recently, the combination between rough sets and matroids has attracted many interesting research. For example, Zhu and Wang [33] established a matroidal structure through the upper approximation number and studied generalized rough sets with matroidal approaches. Liu and Zhu [34] established a parametric matroid through the lower approximation operator of rough sets. Li et al. [35,36] used matroidal approaches to investigate rough sets through closure operators. Su and Zhu [37] presented three types of matroidal structures of coveringbased rough sets. Wang et al. [38] induced a matroid named 2circuit matroid by equivalence relations, and equivalently formulated attribute reduction with matroidal approaches. Wang and Zhu used matrix Symmetry 2018, 10, 418; doi:10.3390/sym10090418
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approaches to study the 2circuit matroid [39], and used contraction operation in matroids to study some relationships between a subset and the upper approximation of this subset in rough sets [40]. Unfortunately, all of these papers never study matroids and rough sets through the positive, negative and boundary operators of rough sets. Thus, it is necessary to further study rough sets and matroids by these operators in this paper. In addition, only Wang et al. [38] presented two equivalent descriptions of attribute reduction by closure operators and rank functions of matroids, respectively. We consider presenting a novel approach to attribute reduction through the girth function of matroids in this paper. In this paper, we mainly use the positive operator, the negative operator and the boundary operator to study matroids and rough sets, and propose a method to compute attribute reduction in information systems through the girth function of matroids. Firstly, we generalize the positive (the lower approximation operator), upper approximation, negative and boundary operators of rough sets to the interior, closure, exterior and boundary operators of matroids respectively. Among them, the upper and lower approximation operators have been studied in [35]. Thus, there are four matroids induced by these four operators of rough sets. Then, the relationship between these four matroids is studied, which implies the relationship about operators between rough sets and matroids. In fact, these four matroids are the same one. Secondly, a relationship about the restriction operation both in matroids and rough sets is proposed. Finally, a matroidal approach is proposed to compute attribute reduction in information systems through the girth function of matroids, and an example about attribute reduction is solved. Using this matroidal approach, we can compute attribute reduction through their results “2” and “∞”. The rest of this paper is organized as follows. Section 2 recalls some basic notions about rough sets, information systems and matroids. In Section 3, we generalize four operators of rough sets to four operators of matroids, respectively. In addition, we study the relationship between four matroids induced by these four operators of rough sets. Moreover, a relationship about operations between matroids and rough sets is presented. In Section 4, an equivalent formulation of attribute reduction through the girth function is presented. Based on the equivalent formulation, a novel method is proposed to compute attribute reduction in information systems. Finally, Section 5 concludes this paper and indicates further works. 2. Basic Deﬁnitions In this section, we review some notions in Pawlak’s rough sets, information systems and matroids. 2.1. Pawlak’s Rough Sets and Information Systems The deﬁnition of approximation operators is presented in [1,41]. Let R an equivalence relation on U. For any X ⊆ U, a pair of approximation R( X ) and R( X ) of X are deﬁned by ? R( X ) = { x ∈ U : RN ( x ) X = ∅}, R( X ) = { x ∈ U : RN ( x ) ⊆ X }, where RN ( x ) = {y ∈ U : xRy}. R and R are called the upper and lower approximation operators with respect to R, respectively. In this paper, U is a nonempty and ﬁnite set called universe. Let − X be the complement of X in U and ∅ be the empty set. We have the following conclusions about R and R. Proposition 1. Refs. [1,41] Let R be an equivalence relation on U. For any X, Y ⊆ U,
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(1L) R(U ) = U,
(1H) R(U ) = U,
(2L) R(φ) = φ,
(2H) R(φ) = φ,
(3L) R( X ) ⊆ X, (4L) R( X
?
Y ) = R( X )
?
(3H) X ⊆ R( X ), R (Y ) ,
(4H) R( X
(5L) R( R( X )) = R( X ),
*
Y ) = R( X )
*
R (Y ) ,
(5H) R( R( X )) = R( X ),
(6L) X ⊆ Y ⇒ R( X ) ⊆ R(Y ),
(6H) X ⊆ Y ⇒ R( X ) ⊆ R(Y ),
(7L) R(− R( X )) = − R( X ),
(7H) R(− R( X )) = − R( X ),
(8LH) R(− X ) = − R( X ),
(9LH) R( X ) ⊆ R( X ).
On the basis of the upper and lower approximation operators with respect to R, one can deﬁne three operators to divide the universe, namely, the negative operator NEGR , the positive operator POSR and the boundary operator BNDR : NEGR ( X ) = U − R( X ), POSR ( X ) = R( X ), BNDR ( X ) = R( X ) − R( X ). An information system [38] is an ordered pair IS = (U, A), where U is a nonempty ﬁnite set of objects and A is a nonempty ﬁnite set of attributes such that a : U → Va for any a ∈ A, where Va is called the value set of a. For all B ⊆ A, the indiscernibility relation induced by B is deﬁned as follows: I ND ( B) = {( x, y) ∈ U × U : ∀b ∈ B, b( x ) = b(y)}. Deﬁnition 1. (Reduct [38]) Let IS = (U, A) be an information system. For all B ⊆ A, B is called a reduct of IS, if the following two conditions hold: (1) I ND ( B) = I ND ( B − b) for any b ∈ B, (2) I ND ( B) = I ND ( A). 2.2. Matroids Deﬁnition 2. (Matroid [29,30]) Let U is a ﬁnite set, and I is a nonempty subset of 2U (the set of all subsets of U). (U, I) is called a matroid, if the following conditions hold: ( I1) If I ∈ I and I ⊆ I, then I ∈ I. * ( I2) If I1 , I2 ∈ I and  I1  <  I2 , then there exists e ∈ I2 − I1 such that I1 {e} ∈ I, where  I  denotes the cardinality of I. Let M = (U, I) be a matroid. We shall often write U ( M ) for U and I( M) for I, particularly when several matroids are being considered. The members of I are the independent sets of M. Example 1. Let U = { a1 , a2 , a3 , a4 , a5 } and I = {∅, { a1 }, { a2 }, { a3 }, { a4 }, { a5 }, { a1 , a3 }, { a1 , a4 }, { a1 , a5 }, { a2 , a3 }, { a2 , a4 }, { a2 , a5 }, { a3 , a4 }, { a3 , a5 }, { a4 , a5 }, { a1 , a3 , a4 }, { a1 , a3 , a5 }, { a1 , a4 , a5 }, { a2 , a3 , a4 }, { a2 , a3 , a5 }, { a2 , a4 , a5 }}. Then, M = (U, I) is a matroid. In order to make some expressions brief, some denotations are presented. Let A ⊆ 2U . Then, Min(A) = { X ∈ A : ∀Y ∈ A, Y ⊆ X ⇒ X = Y }, Max (A) = { X ∈ A : ∀Y ∈ A, X ⊆ Y ⇒ X = Y }, Opp(A) = { X ⊆ U : X ∈ A}.
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The set of all circuits of M is deﬁned as C( M) = Min(Opp(I)). The rank function r M of M is denoted by r M ( X ) = max { I  : I ⊆ X, I ∈ I} for any X ⊆ U. r M ( X ) is called the rank of X in M. The closure operator cl M of M is deﬁned as *
cl M ( X ) = {u ∈ U : r M ( X ) = r M ( X {u})} for all X ⊆ U. We call cl M ( X ) the closure of X in M. X is called a closed set if cl M ( X ) = X, and we denote the family of all closed sets of M by F( M ). The closure axiom of a matroid is introduced in the following proposition. Proposition 2. (Closure axiom [29,30]) Let cl be an operator of U. Then, there exists one and only one matroid M such that cl = cl M iff cl satisﬁes the following four conditions: (CL1) X ⊆ cl ( X ) for any X ⊆ U; (CL2) If X ⊆ Y ⊆ U, then cl ( X ) ⊆ cl (Y ); (CL3) cl (cl ( X )) = cl ( X ) for any X ⊆ U; * * (CL4) For any x, y ∈ U, if y ∈ cl ( X { x }) − cl ( X ), then x ∈ cl ( X {y}). Example 2. (Continued from Example 1) Let X = { a3 , a4 }. Then, C( M ) = Min(Opp(I)) = {{ a1 , a2 }, { a3 , a4 , a5 }}, r M ( X ) = max { I  : I ⊆ X, I ∈ I} = 2, * cl M ( X ) = {u ∈ U : r M ( X ) = r M ( X {u})} = { a3 , a4 , a5 }, F( M) = {∅, { a3 }, { a4 }, { a5 }, { a1 , a2 }, { a1 , a2 , a3 }, { a1 , a2 , a4 }, { a1 , a2 , a5 }, { a3 , a4 , a5 }, { a1 , a2 , a3 , a4 , a5 }}. Based on F( M), the interior operator int M of M is deﬁned as int M ( X ) =
*
{Y ⊆ X : U − Y ∈ F( M)} for any X ⊆ U.
int M ( X ) is called the interior of X in M. X is called a open set if int M ( X ) = X. The following proposition shows the interior axiom of a matroid. Proposition 3. (Interior axiom [29,30]) Let int be an operator of U. Then, there exists one and only one matroid M such that int = int M iff int satisﬁes the following four conditions: ( I NT1) int( X ) ⊆ X for any X ⊆ U, ( I NT2) If X ⊆ Y ⊆ U, then int( X ) ⊆ int(Y ), ( I NT3) int(int( X )) = int( X ) for any X ⊆ U, ( I NT4) For any x, y ∈ U, if y ∈ int( X ) − int( X − { x }), then x ∈ int( X − {y}). Example 3. (Continued from Example 2) int M ( X ) =
*
{Y ⊆ X : U − Y ∈ F( M)} = { a3 , a4 }.
Based on the closure operator cl M , the exterior operator ex M and the boundary operator bo M of M are deﬁned as ex M ( X ) = −cl M ( X ) and bo M ( X ) = cl M ( X )
?
cl M (− X ) for all X ⊆ U.
ex M ( X ) is called the exterior of X in M, and bo M ( X ) is called the boundary of X in M. The following two propositions present the exterior and boundary axioms, respectively. Proposition 4. (Exterior axiom [42]) Let ex be an operator of U. Then, there exists one and only one matroid M such that ex = ex M iff ex M satisﬁes the following four conditions: ? ( EX1) X ex ( X ) = ∅ for any X ⊆ U; ( EX2) If X ⊆ Y ⊆ U, then ex (Y ) ⊆ ex ( X ); ( EX3) ex (−ex ( X )) = ex ( X ) for any X ⊆ U; * * ( EX4) For any x, y ∈ U, if y ∈ ex ( X ) − ex ( X { x }), then x ∈ ex ( X {y}).
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Proposition 5. (Boundary axiom [42]) Let bo be an operator of U. Then, there exists one and only one matroid M such that bo = bo M iff bo satisﬁes the following ﬁve conditions: ( BO1) bo ( X ) = bo (− X ) for any X ⊆ U; ( BO2) bo (bo ( X )) ⊆ bo ( X ) for any X ⊆ U; ? ? * ? ? ? ( BO3) X Y (bo ( X ) bo (Y )) ⊆ X Y bo ( X Y ) for any X, Y ⊆ U; * * ( BO4) For any x, y ∈ U, if y ∈ bo ( X { x }) − bo ( X ), then x ∈ bo ( X {y}); * ( BO5) bo ( X bo ( X )) ⊆ bo ( X ) for any X ⊆ U. Example 4. (Continued from Example 2) ex M ( X ) = U − { a3 , a4 , a5 } = { a1 , a2 }, ? ? bo M ( X ) = cl M ( X ) cl M (− X ) = { a3 , a4 , a5 } { a1 , a2 , a5 } = { a5 }. The following proposition shows some relationships between these above four operators, namely cl M , int M , ex M and bo M . Proposition 6. Ref. [42] Let M = (U, I) be a matroid. For all X ⊆ U, the following statements hold: (1) int M ( X ) = −cl M (− X ) and cl M ( X ) = −int M (− X ); (2) cl M (bo M ( X )) = bo M ( X ); (3) bo M (ex M ( X )) = bo M (− X ). 3. The Relationship about Operators between Rough Sets and Matroids In this section, four matroids are induced by four operators of rough sets. These four matroids are induced by the lower approximation operator R (because R = POSR , we only consider R), the upper approximation operator R, the negative operator NEGR and the boundary operator BNDR through the interior axiom, the closure axiom, the exterior axiom and the boundary axiom, respectively. Among them, the upper approximation operator R has been studied in [35]. Then, the relationship between these four matroids are studied, and we ﬁnd that these four are the same one. According to this work, we present the relationship about operators between rough sets and matroids. 3.1. Four Matroids Induced by Four Operators of Rough Sets In this subsection, we generalize the positive operator (the lower approximation operator), the upper approximation operator, the negative operator and the boundary operator of rough sets to the interior operator, the closure operator, the exterior operator and the boundary operator of matroids, respectively. Firstly, the following lemma is proposed. Lemma 1. Refs. [1,41] Let R be an equivalence relation on U. For any x, y ∈ U, if x ∈ RN (y), then y ∈ RN ( x ). The following proposition shows that the lower approximation operator R satisﬁes the interior axiom of matroids. Proposition 7. Let R be an equivalence relation on U. Then, R satisﬁes ( I NT1), ( I NT2), ( I NT3) and ( I NT4) of Proposition 3. Proof. By (1L), (6L) and (5L) of Proposition 1, R satisﬁes ( I NT1), ( I NT2) and ( I NT3), respectively. ( I NT4): For any x, y ∈ U, if y ∈ R( X ) − R( X − { x }), then y ∈ R( X ) but y ∈ R( X − { x }). Hence, RN (y) ⊆ X but RN (y) ⊆ X − { x }. Therefore, x ∈ RN (y). According to Lemma 1, y ∈ RN ( x ). Hence, RN ( x ) ⊆ X − {y}, i.e., x ∈ R( X − {y}). Inspired by Proposition 7, there is a matroid such that R is its interior operator.
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Deﬁnition 3. Let R be an equivalence relation on U. The matroid whose interior operator is R is denoted by M ( R). We say M( R) is the matroid induced by R. Corollary 1. Let R be an equivalence relation on U. Then, int M( R) = POSR . Proof. According to Deﬁnition 3, int M( R) = R. Since POSR = R, so int M( R) = POSR . The upper approximation operator R satisﬁes the closure axiom in [35,38]. Proposition 8. Refs. [35,38] Let R be an equivalence relation on U. Then, R satisﬁes (CL1), (CL2), (CL3) and (CL4) of Proposition 2. Proposition 8 determines the second matroid induced by R. Deﬁnition 4. Let R be an equivalence relation on U. The matroid whose closure operator is R is denoted by M ( R). We say that M( R) is the matroid induced by R. The negative operator NEGR satisﬁes the exterior axiom. Proposition 9. Let R be an equivalence relation on U. Then, NEGR satisﬁes ( EX1), ( EX2), ( EX3) and ( EX4) of Proposition 4. Proof. ( EX1): For any X ⊆ U, NEGR ( X ) = U − R( X ). According to (3H ) of Proposition 1, X ⊆ R( X ). ? Therefore, X NEGR ( X ) = ∅; ( EX2): According to (6H ) of Proposition 1, if X ⊆ Y ⊆ U, then R( X ) ⊆ R(Y ). Therefore, U − R(Y ) ⊆ U − R(Y ), i.e., NEGR (Y ) ⊆ NEGR ( X ); ( EX3): For any X ⊆ U, NEGR ( X ) = U − R( X ). Hence, − NEGR ( X ) = U − NEGR ( X ) = U − (U − R( X )) = R( X ). Therefore, NEGR (− NEGR ( X )) = NEGR ( R( X )) = U − R( R( X )). According to (5H ) of Proposition 1, R( R( X )) = R( X ). Hence, NEGR (− NEGR ( X )) = U − R( X ) = NEGR ( X ); * ( EX4): For any x, y ∈ U, if y ∈ NEGR ( X ) − NEGR ( X { x }), then y ∈ NEGR ( X ) but y ∈ * * * NEGR ( X { x }), i.e., y ∈ U − R( X ) but y ∈ U − R( X { x }). Since R( X ) ⊆ U and R( X { x }) ⊆ U, * ? * ? so y ∈ R( X { x }) but y ∈ R( X ). Hence, RN (y) ( X { x }) = ∅ but RN (y) X = ∅. Therefore, ? ? * RN (y) { x } = ∅, i.e., x ∈ RN (y). According to Lemma 1, y ∈ RN ( x ). Hence, RN ( x ) ( X {y}) = ∅, * * * i.e., x ∈ R( X {y}). Therefore, x ∈ U − R( X {y}), i.e., x ∈ NEGR ( X {y}). Proposition 9 determines the third matroid such that NEGR is its exterior operator. Deﬁnition 5. Let R be an equivalence relation on U. The matroid whose exterior operator is NEGR is denoted by M( NEGR ). We say M( NEGR ) is the matroid induced by NEGR . In order to certify the boundary operator BNDR satisﬁes the boundary axiom, the following two lemmas are proposed. Lemma 2. Refs. [1,41] Let R be an equivalence relation on U. For all X, Y ⊆ U, R( X Lemma 3. Let R be an equivalence relation on U. If X ⊆ Y ⊆ U, then X ?
?
?
?
?
Y ) ⊆ R( X )
?
R (Y ) .
BNDR (Y ) ⊆ BNDR ( X ). ?
?
Proof. For any x ∈ X BNDR (Y ), X BNDR (Y ) = X ( R( X ) − R( X )) = X R( X ) R(− X )). ? ? Since X ⊆ Y ⊆ U, so −Y ⊆ − X ⊆ U. According to (6H ) of Proposition 1, X R( X ) R(− X )) = ? ? ? X R(− X ) ⊆ R( X ) R(− X ) = BNDR ( X ). Hence, x ∈ BNDR ( X ), i.e., X BNDR (Y ) ⊆ BNDR ( X ). The boundary operator BNDR satisﬁes the boundary axiom.
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Proposition 10. Let R be an equivalence relation on U. Then, BNDR satisﬁes ( BO1), ( BO2), ( BO3), ( BO4) and ( BO5) of Proposition 5. Proof. ( BO1): According to (8LH ) of Proposition 1, R(− X ) = − R( X ). For any X ⊆ U, BNDR (− X )
= R(− X ) − R(− X ) ?
= R(− X ) (U − R(− X )) = (− R( X ))
?
?
R( X )
= R( X ) (− R( X )) = R( X ) − R( X ) = BNDR ( X ). ( BO2): For any X ⊆ U, BNDR ( BNDR ( X ))
= R( BNDR ( X )) − R( BNDR ( X )) ?
= R( BNDR ( X )) (U − R( BNDR ( X ))) ?
= R( BNDR ( X )) (− R( BNDR ( X ))) ?
= R( BNDR ( X )) ( R(− BNDR ( X ))) ⊆ R( BNDR ( X )) = R( R( X ) − R( X )) ?
= R( R( X ) (− R( X ))). According to Lemma 1, we know ?
R( R( X ) (− R( X )))
⊆ R( R( X )) = R( X )
?
?
R(− R( X ))
R(− R( X ))
= R( X ) − R( X ) = BNDR ( X ). Hence, BNDR ( BNDR ( X )) ⊆ BNDR ( X ); ? ? * ? ? * ( BO3): For any X, Y ⊆ U, X Y ( BNDR ( X ) BNDR (Y )) = X Y (( R( X ) − R( X )) ( R(Y ) − ? ? ? * ? ? ? * R(Y ))) = X Y (( R( X ) R(− X )) ( R(Y ) R(−Y ))) ⊆ X Y ( R(− X ) R(−Y )). According ? ? * ? ? * to (4H ) of Proposition 1, we know X Y ( R(− X ) R(−Y )) = X Y R((− X ) (−Y )) = ? ? ? ? ? X Y R(−( X Y )). According to (6H ) of Proposition 1, we know X Y ⊆ R( X Y ). Therefore, ? ? ? ? ? ? ? ? ? ? ? X Y R(−( X Y )) = X Y R(−( X Y )) R( X Y ) = X Y BNDR ( X Y ). ( BO4): When x = y or x ∈ X, it is straightforward. When y ∈ X, it does not hold. (In fact, we suppose * ? * y ∈ X. If y ∈ BNDR ( X { x }), according to Lemma 3, we know y ∈ X BNDR ( X { x }) ⊆ BNDR ( X ),
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Symmetry 2018, 10, 418
*
which is contradictory with y ∈ BNDR ( X { x }) − BNDR ( X ). Hence, y ∈ X.) We only need to prove * it for x = y and x, y ∈ X. If y ∈ BNDR ( X { x }) − BNDR ( X ), since BNDR ( X
*
{ x }) − BNDR ( X )
*
*
= ( R( X { x }) − R( X { x })) − ( R( X ) − R( X )) *
?
*
?
= ( R( X { x }) = ( R( X
*
{ x })
= ( R( X { x }) ?
?
*
*
R(−( X { x }))) − ( R( X ) *
?
R(− X ))
?
*
R(−( X { x }))) ((− R( X )) (− R(− X ))) *
?
*
*
R(−( X { x })) (− R( X ))) ( R( X { x }) ?
( R(−( X { x }))) (− R(− X ))) *
?
*
?
*
=
R( X { x })
=
R( X { x }) (− R( X ))
= ( R( X
*
?
R( −( X { x })) (− R( X ))
{ x }) − R( X ))
?
?
*
R(−( X { x }))
R(−( X
*
*
{ x })),
*
then y ∈ R(X {x}) − R(X) and y ∈ R(−(X {x})). According to Proposition 8, we have x ∈ * * * * ? * R(X {y}). Since y ∈ R(−(X {x})), so x ∈ R(−(X {y})). Hence, y ∈ R(X {y}) R(−(X {y})), * i.e., y ∈ BNDR (X {y}). ( BO5): For any X, Y ⊆ U, BNDR ( X
*
BNDR ( X ))
= R( X = R( X = R( X ⊆ R( X
* * * * *
BNDR ( X )) − R( X BNDR ( X )) BNDR ( X )) BNDR ( X ))
= R( X ( R( X ) = R( R( X )
?
= R( R( X ))
?
U)
?
?
? ? ?
*
R(−( X
BNDR ( X )) *
BNDR ( X )))
?
R((− X ) ( BNDR (− X ))) R(− X )
R(− X )))
?
R(− X )
R(− X )
R(− X ). ?
According to (5H ) and (8LH ) of Proposition 1, R( R( X )) R(− X ) = R( X ) * R( X ) = BNDR ( X ). Therefore, BNDR ( X BNDR ( X )) ⊆ BNDR ( X ).
?
R( − X ) = R( X ) −
Proposition 8 determines the fourth matroid such that BNDR is its boundary operator. Deﬁnition 6. Let R be an equivalence relation on U. The matroid whose boundary operator is BNDR is denoted by M( BNDR ). We say that M ( BNDR ) is the matroid induced by BNDR . 3.2. The Relationship between These Four Matroids This subsection studies the relationship between these four matroids in the above subsection. In fact, these four matroids are the same one. Theorem 1. Let R be an equivalence relation on U. Then, M( R) = M( R) = M( NEGR ) = M( BNDR ). Proof. (1) On one hand, M ( R) and M( R) have the same grand U. On the other hand, according to Deﬁnition 3, we know int M( R) ( X ) = R( X ) for any X ⊆ U. By Proposition 6, cl M( R) ( X ) = −int M( R) (− X ) = − R(− X ). According to (8LH ) of Proposition 1, − R(− X ) = R( X ). Hence, cl M( R) ( X ) = R( X ). According to Deﬁnition 4, cl M( R) ( X ) = R( X ). Therefore, cl M( R) ( X ) = cl M( R) ( X ),
i.e., M( R) = M ( R). (2) On one hand, M( R) and M( NEGR ) have the same grand U. On the other hand, according to Deﬁnition 4, we know cl M( R) = R. For any X ⊆ U, ex M( R) ( X ) = −cl M( R) ( X ) = − R( X ) = U − 367
Symmetry 2018, 10, 418
R( X ) = NEGR ( X ). By Deﬁnition 5, ex M( NEGR ) ( X ) = NEGR ( X ). Hence, ex M( R) ( X ) = ex M( NEGR ) ( X ),
i.e., M( R) = M ( NEGR ). (3) On one hand, M( R) and M( NEGR ) have the same grand U. On the other hand, according ? to Deﬁnition 4, we have cl M( R) = R. For all X ⊆ U, bo M( R) ( X ) = cl M( R) ( X ) cl M( R) (− X ) = ?
?
R( X ) R(− X ) = R( X ) R(− X ) = R( X ) − R( X ) = BNDR ( X ). According to Deﬁnition 6, bo M( NEGR ) ( X ) = BNDR ( X ). Therefore, bo M( R) ( X ) = bo M( NEGR ) ( X ), i.e., M( R) = M ( NEGR ).
Deﬁnition 7. Let R be an equivalence relation on U. The matroid whose interior operator, closure operator, exterior operator and boundary operator are R, R, NEGR and BNDR is deﬁned as M( R). We say that M( R) is the matroid induced by R. According to the above deﬁnition, we have the relationship about operators between rough sets and matroids as Table 1: Table 1. The relationship about operators between rough sets and matroids. M ( R) Is the Matroid Induced by R int M( R) = R = POSR cl M( R) = R ex M( R) = NEGR bo M( R) = BNDR
3.3. The Relationship about Operations between Matroids and Rough Sets In this subection, a relationship about the restriction operation both in matroids and rough sets is proposed. First of all, two deﬁnitions of these two operations are presented in the following two deﬁnitions. Deﬁnition 8. (Restriction [29,30]) Let M = (U, I) be a matroid. For X ⊆ U, the restriction of M to X is deﬁned as M  X = ( X, IX ), where IX = { I ⊆ X : I ∈ I}. Not that C( M X ) = {C ⊆ X : C ∈ C( M )}. For an equivalence relation R on U, there is also a deﬁnition of restriction of R. For any X ⊆ U, R X is an equivalence relation called the restriction of R to X, where R X = {( x, y) ∈ X × X : ( x, y) ∈ R}, X × X is the product set of X and X. According to Deﬁnition 7, M( R X ) is a matroid on X. In [38], the set of independent sets of M ( R) is proposed in the following lemma. Lemma 4. Ref. [38] Let R be an equivalence relation on U. Then, / R }. I( M( R)) = { X ⊆ U : ∀ x, y ∈ X, x = y ⇒ ( x, y) ∈ Example 5. Let R be an equivalence relation on U with U = { a, b, c, d, e}, and U/R = {{ a, b}, {c, d, e}}. According to Lemma 4, I( M( R)) = {∅, { a}, {b}, {c}, {d}, {e}, { a, c}, {b, c}, { a, d}, {b, d}, { a, e}, {b, e}}. Proposition 11. Let R be an equivalence relation on U and X ⊆ U. Then, M( R X ) = M( R) X. Proof. For any X ⊆ U, R X is an equivalence relation on X. Thus, M( R X ) is a matroid on X. By Deﬁnition 8, M ( R) X is a matroid on X. Therefore, we need to prove only I( M( R X )) = I( M( R) X ). According to Lemma 4, I( M( R X )) = {Y ⊆ X : ∀ x, y ∈ Y, x = y ⇒ ( x, y) ∈ / R X }, I( M ( R) X ) = / R}. On one hand, since R X ⊆ R, I( M ( R) X ) ⊆ I( M( R X )). {Y ⊆ X : ∀ x, y ∈ Y, x = y ⇒ ( x, y) ∈ On the other hand, suppose Y ∈ I( M( R X )) − I( M ( R) X ). For any x, y ∈ Y, if x = y, then ( x, y) ∈ / R X but ( x, y) ∈ R. Therefore, x, y ∈ / X but x, y ∈ U, i.e., x, y ∈ U − X. Hence, Y ⊆ U − X, which is contradictory with Y ⊆ X, i.e., Y ∈ I( M( R X )) − I( M ( R) X ). Thus, I( M ( R X )) ⊆ I( M( R) X ). 368
Symmetry 2018, 10, 418
Example 6. (Continued from Example 5) Let X = { a, b, c}. According to Deﬁnition 8, I( M ( R) X ) = {∅, { a}, {b}, {c}, { a, c}, {b, c}}, and M( R) X = ( X, I( M ( R) X )). Since R X = {( a, a), (b, b), (c, c), ( a, b), (b, a)}, so X/( R X ) = {{ a, b}, {c}}. According to Lemma 4, I( M( R X )) = {∅, { a}, {b}, {c}, { a, c}, {b, c}}, and M( R X ) = ( X, I( M( R X )). Therefore, M( R X ) = M( R) X. 4. A Matroidal Approach to Attribute Reduction through the Girth Function In this section, a matroidal approach is proposed to compute attribute reduction in information systems through the girth function of matroids. 4.1. An Equivalent Formulation of Attribute Reduction through the Girth Function Lemma 5. Ref. [15] Let R1 and R2 be two equivalence relations on U, respectively. Then, R1 = R2 if and only if R1 = R2 . Based on Lemma 5, we propose a necessary and sufﬁcient condition for two equivalence relations induce the same matroids. Proposition 12. Let R1 and R2 be two equivalence relations on U, respectively. Then, M ( R1 ) = M( R2 ) if and only if R1 = R2 . Proof. According to Deﬁnition 7, M( R1 ) and M( R2 ) have the same grand U. Proposition 3, Proposition 7 and Lemma 5, M ( R1 ) = M ( R2 )
According to
⇔ int M( R1 ) = int M( R2 ) ⇔ R1 = R2 ⇔ R1 = R2 .
An equivalent formulation of attribute reduction in information systems is presented from the viewpoint of matroids. Proposition 13. Let IS = (U, A) be an information system. For all B ⊆ A, B is a reduct of IS if and only if it satisﬁes the following two conditions: (1) For all b ∈ B, M( I ND ( B)) = M( I ND ( B − b)); (2) M ( I ND ( B)) = M( I ND ( A)). Proof. Since I ND ( A), I ND ( B) and I ND ( B − b) are equivalence relations on U, M ( I ND ( A)), M ( I ND ( B)) and M ( I ND ( B − b)) are matroids on U. According to Proposition 12, (1) For all b ∈ B, M ( I ND ( B)) = M ( I ND ( B − b)) ⇔ I ND ( B) = I ND ( B − b); (2) M ( I ND ( B)) = M( I ND ( A)) ⇔ I ND ( B) = I ND ( A). According to Deﬁnition 1, it is immediate. In Proposition 13, the equivalent formulation of attribute reduction is not convenient for us to compute the attribute reduction. We consider to use the girth function of matroids to compute it. Deﬁnition 9. (Girth function [29,30]) Let M = (U, I) be a matroid. The girth g( M) of M is deﬁned as: ( min{C  : C ∈ C( M)}, C( M) = ∅; g( M) = ∞, C( M) = ∅. For all X ⊆ U, the girth function gM is defined as gM (X) = g( MX). gM (X) is called the girth of X in M.
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Symmetry 2018, 10, 418
According to Deﬁnition 9, the girth function is related to circuits. Thus, the following lemma presents the family of all circuits of M( R). Lemma 6. Ref. [38] Let R be an equivalence relation on U. Then, C( M ( R)) = {{ x, y} ⊆ U : x = y ∧ ( x, y) ∈ R}. Example 7. (Continued from Example 5) C( M( R)) = {{ a, b}, {c, d}, {c, e}, {d, e}}. Based on the characteristics of the matroid induced by an equivalence relation, a type of matroids is abstracted, which is called a 2circuit matroid. M is called a 2circuit matroid if C  = 2 for all C ∈ C( M ). Note that, if C( M ) = ∅, then M is also a 2circuit matroid. In this section, we don’t consider this case. The matroid M ( R) is a 2circuit matroid. Proposition 14. Let R be an equivalence relation on U and X ⊆ U. Then, ( 2, C( M( R)) = ∅; g( M( R)) = ∞, C( M( R)) = ∅; ( g M( R) ( X ) =
C( M ( R) X ) = ∅; C( M ( R) X ) = ∅.
2, ∞,
Proof. Since M( R) is a 2circuit matroid, C  = 2 for all C ∈ C( M ( R)). According to Deﬁnition 9, it is immediate. Corollary 2. Let R be an equivalence relation on U and X ⊆ U. Then, ( 2, ∃ x ∈ U, s.t.,  RN ( x ) ≥ 2; g( M( R)) = ∞, otherwise, (
∃ x ∈ X, s.t.,  RN ( x ) otherwise.
2, ∞,
g M( R) ( X ) =
?
X  ≥ 2;
Proof. According to Lemma 6, C( M ( R)) = ∅
⇔ ∃ x, y ⊆ U, s.t., x = y ∧ ( x, y) ∈ R ⇔ ∃ x ∈ U, s.t.,  RN ( x ) ≥ 2.
Hence,
( g( M ( R)) =
2, ∞,
∃ x ∈ U, s.t.,  RN ( x ) ≥ 2; otherwise.
Since C( M( R) X ) = {C ⊆ X : C ∈ C( M( R))} = {{ x, y} ⊆ X : x = y ∧ ( x, y) ∈ R}, C( M ( R) X ) = ∅
⇔ ∃ x, y ⊆ X, s.t., x = y ∧ ( x, y) ∈ R ⇔ ∃ x ∈ U, s.t.,  RN ( x )
Hence,
( g M( R) ( X ) =
2, ∞,
?
∃ x ∈ X, s.t.,  RN ( x ) otherwise.
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X  ≥ 2.
?
X  ≥ 2;
Symmetry 2018, 10, 418
Lemma 7. Refs. [1,41] Let R1 and R2 be two equivalence relations on U, respectively. Then, for any x ∈ U,
( R1
?
R2 ) N ( x ) = R1 N ( x )
?
R2 N ( x ).
According to Corollary 2, the girth function of the matroid induced by attribute subsets is presented in the following proposition. Proposition 15. Let IS = (U, A) be an information system and X ⊆ U. Then, for all B ⊆ A, ⎧ ? ⎨ 2, ∃ x ∈ U, s.t.,  Ri N ( x ) ≥ 2; R g( M( I ND ( B))) = i ∈B ⎩ ∞, otherwise, g M( I ND( B)) ( X ) =
⎧ ⎨ 2,
∃ x ∈ X, s.t., (
⎩ ∞,
otherwise.
? Ri ∈ B
Ri N ( x ))
?
X  ≥ 2;
Proof. According to Lemma 7 and Corollary 2, it is immediate. Note that Ri in Ri N denotes the equivalence relation induced by attribute Ri ∈ A. According to the girth axiom, we know that a matroid is corresponding to one and only one girth function. *
Proposition 16. (Girth axiom [29,30]) Let g : 2U → Z+ {0, ∞} be a function. Then, there exists one and only one matroid M such that g = g M iff g satisﬁes the following three conditions: ( G1) If X ⊆ U and g( X ) < ∞, then X has a subset Y such that g( X ) = g(Y ) = Y . ( G2) If X ⊆ Y ⊆ U, then g( X ) ≥ g(Y ). * ( G3) If X and Y are distinct subsets of U with g( X ) =  X , g(Y ) = Y , then g(( X Y ) − {e}) < ∞ for any ? e ∈ X Y. Inspired by Propositions 13 and 16, we can use the girth function in matroids to compute attribute reduction. Theorem 2. Let IS = (U, A) be an information system. For all B ⊆ A, B is a reduct of IS if and only if it satisﬁes the following two conditions: (1) For all b ∈ B, there exists X ⊆ U such that g M( I ND( B)) ( X ) = g M( I ND( B−b)) ( X ). (2) For all X ⊆ U, g M( I ND( B)) ( X ) = g M( I ND( A)) ( X ). Proof. According to Propositions 13 and 16, it is immediate. 4.2. The Process of the Matroidal Methodology In this subsection, we give the process of the matroidal approach to compute attribute reduction in information systems according to the equivalent description in Section 4.1. In order to obtain all results of an information system IS = (U, A), we need to compute g M( I ND( B)) ( X ) for all B ⊆ A and X ⊆ U based on Theorem 2. According to Deﬁnition 1, we know a reduct of IS will not be ∅. Hence, we only consider B ⊆ A and B = ∅. On the other hand, for all X ⊆ U and B ⊆ A, if  X  ≤ 1, then g M( I ND( B)) ( X ) = g M( I ND( A)) ( X ). According to Theorem 2, we only consider X whose  X  ≥ 2. Therefore, the process is shown as follows: • • •
Input: An information system IS = (U, A), where U = {u1 , u2 , · · · , un } and A = { a1 , a2 , · · · , a m }. Output: All results of IS. Step 1: Suppose Bi ⊂ A (Bi = ∅ and i = 1, 2, · · · , 2m − 2), we compute all I ND ( Bi ) and I ND ( A).
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Symmetry 2018, 10, 418
• •
Step 2: For any i = 1, 2, · · · , 2m − 2, we compute g M( I ND( Bi )) ( X ) and g M( I ND( A)) ( X ) for any X ⊆ U and  X  ≥ 2. Step 3: Obtain all results of IS according to Theorem 2.
4.3. An Applied Example Example 8. Let us consider the following information system IS = (U, A) as is shown in Table 2. Table 2. An information system.
u1 u2 u3 u4 u5
a1
a2
a3
0 1 1 2 1
1 2 0 1 1
0 2 0 1 2
Let B1 = { a1 }, B2 = { a2 }, B3 = { a3 }, B4 = { a1 , a2 }, B5 = { a1 , a3 }, B6 = { a2 , a3 }, A = { a1 , a1 , a3 }. gBi denotes g M( I ND( Bi )) for 1 ≤ i ≤ 6 and g A denotes g M( I ND( A)) . All girth functions induced by attribute subsets as is shown in Table 3. Table 3. Girth functions induced by attribute subsets.
u1 , u2 u1 , u3 u1 , u4 u1 , u5 u2 , u3 u2 , u4 u2 , u5 u3 , u4 u3 , u5 u4 , u5 u1 , u2 , u3 u1 , u2 , u4 u1 , u2 , u5 u1 , u3 , u4 u1 , u3 , u5 u1 , u4 , u5 u2 , u3 , u4 u2 , u3 , u5 u2 , u4 , u5 u3 , u4 , u5 u1 , u2 , u3 , u4 u1 , u2 , u3 , u5 u1 , u2 , u4 , u5 u1 , u3 , u4 , u5 u2 , u3 , u4 , u5 u1 , u2 , u3 , u4 , u5
g B1
g B2
g B3
g B4
g B5
g B6
gA
∞ ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ ∞ 2 2 ∞ ∞ 2 2 ∞ ∞ 2 2
∞ ∞ 2 2 ∞ ∞ ∞ ∞ ∞ 2 ∞ 2 2 2 2 2 ∞ ∞ 2 2 2 2 2 2 2 2
∞ 2 ∞ ∞ ∞ ∞ 2 ∞ ∞ ∞ 2 ∞ 2 2 2 ∞ ∞ ∞ 2 ∞ 2 2 2 2 2 2
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
∞ ∞ ∞ ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ 2 2 ∞ ∞ 2 2 ∞ 2 2
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Accordingly, there are two reducts of IS: B4 = { a1 , a2 } and B6 = { a2 , a3 }. 5. Conclusions In this paper, we generalize four operators of rough sets to four operators of matroids through the interior axiom, the closure axiom, the exterior axiom and the boundary axiom, respectively. Moreover,
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Symmetry 2018, 10, 418
we present a matroidal approach to compute attribute reduction in information systems. The main conclusions in this paper and the continuous work to do are listed as follows: 1.
2.
3.
There are four matroids induced by these four operators of rough sets. In fact, these four matroids are the same one, which implies the relationship about operators between rough sets and matroids. In this work, we assume an equivalence relation. However, there are other structures have been used in rough set theory, among them, tolerance relations [43], similarity relations [44], and binary relations [15,45]. Hence, they can suggest as a future research, the possibility of extending their ideas to these types of settings. The girth function of matroids is used to compute attribute reduction in information systems. This work can be viewed as a bridge linking matroids and information systems in the theoretical impact. In the practical impact, it is a novel method by which calculations will become algorithmic and can be implemented by a computer. Based on this work, we can use the girth function of matroids for attribute reduction in decision systems in the future. In the future, we will further expand the research content of this paper based on some new studies on neutrosophic sets and related algebraic structures [46–50].
Author Contributions: This paper is written through contributions of all authors. The individual contributions and responsibilities of all authors can be described as follows: the idea of this whole thesis was put forward by X.Z., and he also completed the preparatory work of the paper. J.W. analyzed the existing work of rough sets and matroids, and wrote the paper. Funding: This research is funded by the National Natural Science Foundation of China under Grant Nos. 61573240 and 61473239. Conﬂicts of Interest: The authors declare no conﬂicts of interest.
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375
SS symmetry Article
Some Results on Multigranulation Neutrosophic Rough Sets on a Single Domain Hu Zhao 1, * and HongYing Zhang 2 1 2
*
School of Science, Xi’an Polytechnic University, Xi’an 710048, China School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China; [email protected] Correspondence: [email protected] or [email protected]; Tel.: +8618702942326
Received: 11 August 2018; Accepted: 13 September 2018; Published: 19 September 2018
Abstract: As a generalization of single value neutrosophic rough sets, the concept of multigranulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the onedimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved. Keywords: neutrosophic set; neutrosophic rough set; pessimistic (optimistic) multigranulation neutrosophic approximation operators; complete lattice
1. Introduction In order to deal with imprecise information and inconsistent knowledge, Smarandache [1,2] ﬁrst introduced the notion of neutrosophic set by fusing a tricomponent set and the nonstandard analysis. A neutrosophic set consists of three membership functions, where every function value is a real standard or nonstandard subset of the nonstandard unit interval ]0− , 1+ [. Since then, many authors have studied various aspects of neutrosophic sets from different points of view, for example, in order to apply the neutrosophic idea to logics, Rivieccio [3] proposed neutrosophic logics which is a generalization of fuzzy logics and studied some basic properties. Guo and Cheng [4] and Guo and Sengur [5] obtained good applications in cluster analysis and image processing by using neutrosophic sets. Salama and Broumi [6] and Broumi and Smarandache [7] ﬁrst introduced the concept of rough neutrosophic sets, handled incomplete and indeterminate information, and studied some operations and their properties. In order to apply neutrosophic sets conveniently, Wang et al. [8] proposed single valued neutrosophic sets by simplifying neutrosophic sets. Single valued neutrosophic sets can also be viewed as a generalization of intuitionistic fuzzy sets (Atanassov [9]). Single valued neutrosophic sets have become a new majorly research issue. Ye [10–12] proposed decision making based on correlation coefﬁcients and weighted correlation coefﬁcient of single valued neutrosophic sets, and gave an application of proposed methods. Majumdar and Samant [13] studied similarity, distance and entropy of single valued neutrosophic sets from a theoretical aspect. S¸ ahin and Küçük [14] gave a subsethood measure of single valued neutrosophic sets based on distance and showed its effectiveness through an example. We know that there’s a certain connection among fuzzy rough approximation operators and fuzzy relations (resp., fuzzy topologies, information systems [15–17]). Hence, Yang et al. [18] ﬁrstly proposed neutrosophic relations and studied some Symmetry 2018, 10, 417; doi:10.3390/sym10090417
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kinds of kernels and closures of neutrosophic relations. Subsequently they proposed single valued neutrosophic rough sets [19] by fusing single valued neutrosophic sets and rough sets (Pawlak, [20]), and they studied some properties of single value neutrosophic upper and lower approximation operators. As a generalization of single value neutrosophic rough sets, Bao and Yang [21] introduced pdimension single valued neutrosophic reﬁned rough sets, and they also gave some properties of pdimension single valued neutrosophic upper and lower approximation operators. As another generalization of single value neutrosophic rough sets, Bo et al. [22] proposed the concept of multigranulation neutrosophic rough sets and obtained some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. However, the lattice structures of those rough approximation operators in references [19,21,22], were not well studied. Following this idea, Zhao and Zhang [23] gave the supremum and inﬁmum of the pdimension neutrosophic upper and lower approximation operators, but they did not study the relationship between the pdimension neutrosophic upper approximation operators and the pdimension neutrosophic lower approximation operators, especially in the onedimensional case. Inspired by paper [23], a natural problem is: Can the lattice structure of pessimistic (optimistic) multigranulation neutrosophic approximation operators be given? In the present paper, we study the algebraic structure of optimistic (pessimistic) multigranulation single valued neutrosophic approximation operators. The structure of the paper is organized as follows. The next section reviews some basic definitions of neutrosophic sets and onedimensional multigranulation rough sets. In Section 3, the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators are studied. In Section 4, for special neutrosophic relations, a onetoone correspondence relationship between neutrosophic upper approximation operators and lower approximation operators is given. Finally, Section 5 concludes this article and points out the deﬁciencies of the current research. 2. Preliminaries In this section, we brieﬂy recall several deﬁnitions of neutrosophic set (here “neutrosophic set” refers exclusively to “single value neutrosophic set”) and onedimensional multigranulation rough set. Deﬁnition 1 ([8]). A neutrosophic set B in X is deﬁned as follows: ∀ a ∈ X, B = ( TA ( a), I A ( a), FA ( a)), where TA ( a), I A ( a), FA ( a) ∈ [0, 1], 0 ≤ supTA ( a) + supI A ( a) + supFA ( a) ≤ 3. The set of all neutrosophic sets on X will be denoted by SVNS( X ). Deﬁnition 2 ([11]). Let C and D be two neutrosophic sets in X, if TC ( a) ≤ TD ( a), IC ( a) ≥ ID ( a) and FC ( a) ≥ FD ( a) for each a ∈ X, then we called C is contained in D, i.e., C D. If C D and D C, then we called C is equal to D, denoted by C = D. Deﬁnition 3 ([18]). Let A and B be two neutrosophic sets in X, (1)
The union of A and B is a s neutrosophic set C, denoted by A B, where ∀ x ∈ X, TC ( a) = max { TA ( a), TB ( a)}, IC ( a) = min{ I A ( a), IB ( a)}, and FC ( a) = min{ FA ( a), FB ( a)}.
377
Symmetry 2018, 10, 417
(2)
The intersection of A and B is a neutrosophic set D, denoted by A B, where ∀ x ∈ X, TD ( a) = min{ TA ( a), TB ( a)}, ID ( a) = max { I A ( a), IB ( a)}, and FD ( a) = max { FA ( a), FB ( a)}.
Deﬁnition 4 ([18]). A neutrosophic relation R in X is deﬁned as follows: R = {< ( a, b), TR ( a, b), IR ( a, b), FR ( a, b) > ( a, b) ∈ X × X }, where TR : X × X → [0, 1], IR : X × X → [0, 1], FR : X × X → [0, 1] , and 0 ≤ supTR ( a, b) + supIR ( a, b) + supFR ( a, b) ≤ 3. The family of all neutrosophic relations in X will be denoted by SVNR( X ), and the pair ( X, R) is called a neutrosophic approximation space. Deﬁnition 5 ([19]). Let ( X, R) be a neutrosophic approximation space, ∀ A ∈ SVNS( X ), the lower and upper approximations of A with respect to ( X, R), denoted by R( A) and R( A), are two neutrosophic sets whose membership functions are deﬁned as: ∀ a ∈ X, TR( A) ( a) = ∧ [ FR ( a, b) ∨ TA (b)], IR( A) ( a) = ∨ [(1 − IR ( a, b)) ∧ I A (b)], b∈ X
b∈ X
FR( A) ( a) = ∨ [ TR ( a, b) ∧ FA (b)], TR( A) ( a) = ∨ [ TR ( a, b) ∧ TA (b)], b∈ X
b∈ X
IR( A) ( a) = ∧ [ IR ( a, b) ∨ I A (b)], FR( A) ( a) = ∧ [ FR ( a, b) ∨ FA (b)]. b∈ X
b∈ X
The pair ( R( A), R( A)) is called the onedimensional multigranulation rough set (also called single value neutrosophic rough set or onedimension single valued neutrosophic reﬁned rough set) of A with respect to ( X, R). R and R are referred to as the neutrosophic lower and upper approximation operators,respectively. Lemma 1 ([19]). Let R1 and R2 be two neutrosophic relations in X, ∀ A ∈ SVNS( X ), we have (1) (2) (3) (4)
R1 R2 ( A ) = R1 ( A ) R2 ( A ); R1 R2 ( A ) = R1 ( A ) R2 ( A ); R1 R2 ( A ) R1 ( A ) R2 ( A ) R1 ( A ) R2 ( A ); R1 R2 ( A ) R1 ( A ) R2 ( A ).
3. The Lattice Structure of the Pessimistic Multigranulation Neutrosophic Rough Approximation Operators In this section, set M = { R1 , R2 , · · · , Rn } = { Ri }i=1,n is called a multigranulation neutrosophic relations set on X if each Ri is a neutrosophic relation on X. In this case, the pair ( X, M ) will be called an ndimensional multigranulation neutrosophic apptoximation space. Deﬁnition 6 ([22]). Let ( X, M) be an ndimensional multigranulation neutrosophic apptoximation space. We deﬁne two pairs of approximation operators as follows, for all ∀ A ∈ SVNS( X ) and a ∈ X, O
P
MO ( A) = ( MO ( A), M ( A)), M P ( A) = ( M P ( A), M ( A)), where TMO ( A) ( a) = ∨in=1 TRi ( A) ( a), I MO ( A) ( a) = ∧in=1 IRi ( A) ( a), FMO ( A) ( a) = ∧in=1 FRi ( A) ( a). T
O
M ( A)
( a) = ∧in=1 TRi ( A) ( a), I
O
M ( A)
( a) = ∨in=1 IRi ( A) ( a), F
O
M ( A)
378
( a) = ∨in=1 FRi ( A) ( a).
Symmetry 2018, 10, 417
TM P ( A) ( a) = ∧in=1 TRi ( A) ( a), I M P ( A) ( a) = ∨in=1 IRi ( A) ( a), FM P ( A) ( a) = ∨in=1 FRi ( A) ( a). T
P
M ( A)
( a) = ∨in=1 TRi ( A) ( a), I M P ( A) ( a) = ∧in=1 IRi ( A) ( a), FM P ( A) ( a) = ∧in=1 FRi ( A) ( a). O
Then the pair MO ( A) = ( MO ( A), M ( A)) is called an optismistic multigranulation neutrosophic rough P set, and the pair M P ( A) = ( M P ( A), M ( A)) is called an pessimistic multigranulation neutrosophic rough set O
P
M and M are referred to as the optimistic and pessimistic multigranulation neutrosophic upper approximation operators, respectively. Similarly, MO and M P are referred to as the optimistic and pessimistic multigranulation neutrosophic lower approximation operators, respectively. Remark 1. If n = 1, then the multigranulation neutrosophic rough set will degenerated to a onedimensional multigranulation rough set (see Deﬁnition 5). In the following, the family of all multigranulation neutrosophic relations set on X will be denoted by n − SVNR( X ). Deﬁned a relation ) on n − SVNR( X ) as follows: M ) N if and only if Mi Ni , then (n − SVNR( X ), )) is a poset, where M = { Mi }i=1,n and N = { Ni }i=1,n . # $ 3 4 j ∀ M j j∈Λ ⊆ n − SVNR( X ), where M j = Mi and Λ be a index set, we can deﬁne union and i =1,n
intersection of M j as follows: # $ j ∨ M j = j ∈ Λ Mi
j∈Λ
where T
j
j ∈ Λ Mi
F
j∈Λ
j∈Λ
∨
j∈Λ
Mj
and ∧
j∈Λ
and ∧
Mj
( a, b) = ∨ TM j ( a, b), I
i
j
j ∈ Λ Mi
Then ∨
j∈Λ
j∈Λ
i
j∈Λ
i
j ( a, b ) = ∧ F j ( a, b ), T M M
I Mj
i =1,n
# $ j , ∧ M j = j ∈ Λ Mi
( a, b) = ∨ IM j ( a, b), F j∈Λ
i
j j ∈ Λ Mi
j∈Λ
i =1,n
,
( a, b) = ∧ IM j ( a, b), j∈Λ
i
j ( a, b ) = ∧ T j ( a, b ), M M i
j
j ∈ Λ Mi
j∈Λ
i
j∈Λ
i
( a, b) = ∨ FM j ( a, b).
Mj
are two multigranulation neutrosophic relations on X, and we easily show that 3 4 are inﬁmum and supremum of M j j∈Λ , respectively. Hence we can easily obtain the j∈Λ
following theorem: Theorem 1.
@n = { Xn , Xn , · · · , Xn } and ∅ A (n − SVNR( X ), ), ∧, ∨) is a complete lattice, X N = 9 :7 8 n
{∅ N , ∅ N , · · · , ∅ N } are its top element and bottom element, respectively, where Xn and ∅ N are two 9 :7 8 n
neutrosophic relations in X and deﬁned as follows: ∀( a, b) ∈ X × X, TX N ( a, b) = 1, IX N ( a, b) = 0, FX N ( a, b) = 0 and T∅N ( a, b) = 0, I∅N ( a, b) = 1, F∅N ( a, b) = 1. In particular, (SVNR( X ), , , ) is a complete lattice. Theorem 2. Let M = { Ri }i=1,n and N = { Qi }i=1,n be two multigranulation neutrosophic relations set on X, ∀ A ∈ SVNS( X ), we have (1)
M ∨ N O ( A ) MO ( A ) N O ( A ) , M ∨ N P ( A ) = M P ( A ) N P ( A ) ;
(2) (3)
M ∨ N ( A ) M ( A ) N ( A ), M ∨ N ( A ) = M ( A ) N ( A ); M ∧ N O ( A ) MO ( A ) N O ( A ) MO ( A ) N O ( A ) , M ∧ N P ( A ) M P ( A ) N P ( A ) M P ( A ) N P ( A );
(4)
M ∧ N ( A ) M ( A ) N ( A ), M ∧ N ( A ) M ( A ) N ( A ).
O
O
O
P
P
P
O
O
O
P
P
P
379
Symmetry 2018, 10, 417
Proof. We only show that the case of the optimistic multigranulation neutrosophic approximation operators. (1)
∀ a ∈ X, by Lemma 1 and Deﬁnition 6, we have the following: TM∨ NO ( A) ( a) = ∨in=1 TRi Qi ( A) ( a) = ∨in=1 TRi ( A)Qi ( A) ( a) 0 1 = ∨in=1 TRi ( A) ( a) ∧ TQi ( A) ( a) 0 1 0 1 ≤ ∨in=1 TRi ( A) ( a) ∧ ∨in=1 TQi ( A) ( a)
= TMO ( A) ( a) ∧ TNO ( A) ( a) = TMO ( A) NO ( A) ( a),
I M∨ N O ( A) ( a ) = ∧in=1 IRi Qi ( A) ( a) = ∧in=1 IRi ( A)Qi ( A) ( a) 0 1 = ∧in=1 IRi ( A) ( a) ∨ IQi ( A) ( a) 1 0 1 0 ≥ ∧in=1 IRi ( A) ( a) ∨ ∧in=1 IQi ( A) ( a)
= I MO ( A ) ( a ) ∨ I N O ( A ) ( a ) = IMO ( A) NO ( A) ( a),
FM∨ NO ( A) ( a) = ∧in=1 FRi Qi ( A) ( a) = ∧in=1 FRi ( A)Qi ( A) ( a) 0 1 = ∧in=1 FRi ( A) ( a) ∨ FQi ( A) ( a) 0 1 0 1 ≥ ∧in=1 FRi ( A) ( a) ∨ ∧in=1 FQi ( A) ( a)
= FMO ( A) ( a) ∨ FNO ( A) ( a) = FMO ( A) NO ( A) ( a).
(2)
Hence, M ∨ N O ( A) MO ( A) N O ( A). ∀ a ∈ X, by Lemma 1 and Deﬁnition 6, we have the following: T
O
( a)
O
( a) ∨ T
M∨ N ( A) = ∧in=1 TRi Qi ( A) ( a) = ∧in=1 TRi ( A)Qi ( A) ( a) 0 1 = ∧in=1 TRi ( A) ( a) ∨ TQi ( A) ( a) 0 1 0 1 ≥ ∧in=1 TRi ( A) ( a) ∨ ∧in=1 TQi ( A) ( a)
=T
M ( A)
I
O
O
N ( A)
M∨ N ( A)
( a) = T
O
O
M ( A) N ( A)
( a ),
( a)
= ∨in=1 IRi Qi ( A) ( a) = ∨in=1 IRi ( A)Qi ( A) ( a) 0 1 = ∨in=1 IRi ( A) ( a) ∧ IQi ( A) ( a) 0 1 0 1 ≤ ∨in=1 IRi ( A) ( a) ∧ ∨in=1 IQi ( A) ( a) =I
F
O
( a) ∧ I
O
( a)
O
( a) ∧ F
M ( A)
O
N ( A)
( a) = I
O
O
M ( A) N ( A)
( a ),
M∨ N ( A) = ∨in=1 FRi Qi ( A) ( a) = ∨in=1 FRi ( A)Qi ( A) ( a) 0 1 = ∨in=1 FRi ( A) ( a) ∧ FQi ( A) ( a) 0 1 0 1 ≤ ∨in=1 FRi ( A) ( a) ∧ ∨in=1 FQi ( A) ( a)
=F
M ( A)
O
O
O
N ( A)
O
Hence, M ∨ N ( A) M ( A) N ( A). 380
( a) = F
O
O
M ( A) N ( A)
( a ).
Symmetry 2018, 10, 417
(3)
∀ a ∈ X, by Lemma 1 and Deﬁnition 6, we have the following: TM∧ NO ( A) ( a) = ∨in=1 TRi Qi ( A) ( a) ≥ ∨in=1 TRi ( A)Qi ( A) ( a) 0 1 = ∨in=1 TRi ( A) ( a) ∨ TQi ( A) ( a) 0 1 0 1 = ∨in=1 TRi ( A) ( a) ∨ ∨in=1 TQi ( A) ( a)
= TMO ( A) ( a) ∨ TNO ( A) ( a) ≥ TMO ( A) ( a) ∧ TNO ( A) ( a), I M∧ N O ( A) ( a ) = ∧in=1 IRi Qi ( A) ( a) ≤ ∧in=1 IRi ( A)Qi ( A) ( a) 0 1 = ∧in=1 IRi ( A) ( a) ∧ IQi ( A) ( a) 0 1 0 1 = ∧in=1 IRi ( A) ( a) ∧ ∧in=1 IQi ( A) ( a)
= I MO ( A ) ( a ) ∧ I N O ( A ) ( a ) ≤ I MO ( A ) ( a ) ∨ I N O ( A ) ( a ) ,
FM∧ NO ( A) ( a) = ∧in=1 FRi Qi ( A) ( a) ≤ ∧in=1 FRi ( A)Qi ( A) ( a) 0 1 = ∧in=1 FRi ( A) ( a) ∧ FQi ( A) ( a) 0 1 0 1 = ∧in=1 FRi ( A) ( a) ∧ ∧in=1 FQi ( A) ( a)
= FMO ( A) ( a) ∧ FNO ( A) ( a) ≤ FMO ( A) ( a) ∨ FNO ( A) ( a).
(4)
Hence, M ∧ N o ( A) Mo ( A) N o ( A) Mo ( A) N o ( A). ∀ a ∈ X, by Lemma 1 and Deﬁnition 6, we have the following: T
( a)
O
M∧ N ( A)
= ∧in=1 TRi Qi ( A) ( a) ≤ ∧in=1 TRi ( A)Qi ( A) ( a) 0 1 = ∧in=1 TRi ( A) ( a) ∧ TQi ( A) ( a) 0 1 0 1 = ∧in=1 TRi ( A) ( a) ∧ ∧in=1 TQi ( A) ( a) =T
O
( a) ∧ T
O
( a)
O
( a) ∨ T
O
( a)
M ( A)
I
O
N ( A)
( a) = T
O
O
M ( A) N ( A)
M∧ N ( A) = ∨in=1 IRi Qi ( A) ( a) ≥ ∨in=1 IRi ( A)Qi ( A) ( a) 0 1 = ∨in=1 IRi ( A) ( a) ∨ IQi ( A) ( a) 0 1 0 1 = ∨in=1 IRi ( A) ( a) ∨ ∨in=1 IQi ( A) ( a)
=I
M ( A)
F
O
N ( A)
M∧ N ( A)
( a) = I
O
O
M ( A) N ( A)
( a ),
= ∨in=1 FRi Qi ( A) ( a) ≥ ∨in=1 FRi ( A)Qi ( A) ( a) 0 1 = ∨in=1 FRi ( A) ( a) ∨ FQi ( A) ( a) 0 1 0 1 = ∨in=1 FRi ( A) ( a) ∨ ∨in=1 FQi ( A) ( a) =F
O
M ( A)
O
O
( a) ∨ F
O
N ( A)
( a) = F
O
M ( A) N ( A)
O
Hence, M ∧ N ( A) M ( A) N ( A). From Theorem 2, we can easily obtain the following corollary:
381
O
( a ),
( a ).
Symmetry 2018, 10, 417
Corollary 1. Let M = { Ri }i=1,n and N = { Qi }i=1,n be two multigranulation neutrosophic relations set on X. If M ) N, then ∀ A ∈ SVNS( X ), O
O
P
P
N O ( A) MO ( A), N P ( A) M P ( A)), M ( A) N ( A), M ( A) N ( A). # $ # $ P Let HnP = M  M ∈ n − SVNR( X ) and LnP = M P  M ∈ n − SVNR( X ) be the set of pessimistic multigranulation neutrosophic upper and lower approximation operators in X, respectively. ˆ on HnP as follows: M P ≤ ˆ N P if and only if M P ( A) N P ( A) for each Deﬁned a relation ≤ P ˆ A ∈ SVNS( X ). Then ( Hn , ≤) is a poset. ˆ on LnP as follows: M P ≤ ˆ N P if and only if N P ( A) M P ( A) for each A ∈ • Deﬁned a relation ≤ ˆ is a poset. SVNS( X ). Then ( LnP , ≤) # $ # $ O O Let HnO = M  M ∈ n − SVNR( X ) and LO n = M  M ∈ n − SVNR( X ) be the set of optimistic multigranulation neutrosophic upper and lower approximation operators in X, respectively.
•
ˆ on HnO as follows: MO ≤ ˆ N O if and only if MO ( A) N O ( A) for each Deﬁned a relation ≤ ˆ is a poset. A ∈ SVNS( X ). Then ( HnO , ≤) O ˆ O O O ˆ on LO • Deﬁned a relation ≤ n as follows: M ≤ N if and only if N ( A )) M ( A ) for each O ˆ is a poset. A ∈ SVNS( X ). Then ( Ln , ≤) # $ P ˆ and I be a index set, we can deﬁne union and intersection of MiP ⊆( HnP , ≤) Theorem 3. (1) ∀ Mi
•
i∈ I
as follows:
P
P P P ∨ˆ Mi = ∨ Mi , ∧ˆ Mi = [ ∧ Mi ] ,
i∈ I
i∈ I
i∈ I
i∈ I
# $ P P ˆ MiP and where [ ∧ Mi ] = ∨ M ∈ n − SVNR( X )  ∀ A ∈ SVNS( X ), M ( A) i∈ I Mi ( A) . Then ∨ i∈ I i∈ I # $ P P ∧ˆ Mi are supremum and inﬁmum of Mi , respectively. i∈ I i∈ I # $ ˆ and I be a index set, we can deﬁne union and intersection of MiP as follows: ⊆( LnP , ≤) (2) ∀ MiP i∈ I
∨ˆ MiP = ∨ Mi P , ∧ˆ MiP = [ ∨ Mi ] P ,
i∈ I
i∈ I
i∈ I
i∈ I
#
$ ˆ MiP and where [ ∨ Mi ] = ∨ M ∈ n − SVNR( X )  ∀ A ∈ SVNS( X ), i∈ I MiP ( A) M P ( A) . Then ∨ i∈ I i∈ I # $ ∧ˆ MiP are supremum and inﬁmum of MiP , respectively. i∈ I
i∈ I
Proof. We only show (1). P
P
Let M = ∨ Mi , then Mi ) M for each i ∈ I. By Corollary 1, we have Mi ( A) M ( A) for i∈ I
P
P
ˆ M . If M is a multigranulation neutrosophic relations set such that any A ∈ SVNS( X ). Thus Mi ≤ P ˆ M P for each i ∈ I, then A ∈ SVNS( X ), Mi P ( A) M P ( A). Hence, Mi ≤ P
P
P
P
M ( A ) = ∨ Mi ( A ) = i ∈ I Mi ( A ) M ( A ) . i∈ I
$ # P ˆ M P . So ∨ ˆ MiP = ∨ Mi P is the supremum of MiP Thus M ≤ i∈ I
i∈ I
Let Q = [ ∧ Mi ], then ∀ B ∈ SVNS( X ), we have
i∈ I
.
i∈ I
P
P
P
P
Q ( B ) = [ ∧ Mi ] ( B ) i ∈ I M i ( B ) M i ( B ) . i∈ I
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P ˆ MiP for each i ∈ I. If M∗ is a multigranulation neutrosophic relations set such that Thus Q ≤
P ˆ MiP M∗ ≤
for each i ∈ I, then
P
P
M ∗ ( A ) i ∈ I M i ( A ) .
By the construction of [ ∧ Mi ], we can easily obtain M∗ ) [ ∧ Mi ] = Q. Hence, i∈ I
i∈ I
P ˆ [ ∧ Mi ] P = Q P , M∗ ≤ i∈ I
#
P
ˆ MiP = [ ∧ Mi ] is the inﬁmum of MiP So ∧ i∈ I
$ i∈ I
i∈ I
.
Remark 2. (1) ∀ A ∈ SVNS( X ), ∀a ∈ X, we can calculate that the following formula holds. T
P
A ∅ N ( A)
TA P
∅ N ( A)
( a) = 0, I
P
A ∅ N ( A)
( a) = 1, F
P
A ∅ N ( A)
( a) = 1,
( a) = 1, I∅ ( a) = 0, F∅ ( a) = 0. A P ( A) A P ( A) N
N
P
P
P P P ˆ MP A A A Hence, ∀ M ∈ n − SVNR( X ), ∅ N ( A ). It shows that ∅ N ( A ) M ( A ) and M ( A ) ∅ N ≤ P
P P P ˆ and ∅ P ˆ . By ˆ M P , i.e., ∅ A A A and ∅ N ≤ N is the bottom element of ( Hn , ≤) N is the bottom element of ( Ln , ≤) ˆ ,∧ ˆ ,∧ ˆ , ∨) ˆ and ( LnP , ≤ ˆ , ∨) ˆ are complete lattices. Theorem 3, we have the following result: Both ( HnP , ≤ ˆ ,∧ ˆ ˆ , ∨) ˆ , ∨) ˆ and ( LO ˆ are complete lattices if we can use (2) Similarly, we can prove that both ( HnO , ≤ n , ≤, ∧ the generalization formula of O
O
O
M ∨ N ( A) M ( A) N ( A) and M ∨ N O ( A) MO ( A) N O ( A), However, by Theorem 2, we known that O
O
O
M ∨ N ( A) M ( A) N ( A) and M ∨ N O ( A) MO ( A) N O ( A). So, naturally, there is the following problem: How to give the supremum and inﬁmum of the optimistic multigranulation neutrosophic rough approximation operators? 3 4 In the onedimensional case, for convenience, we will use H = R  R ∈ SVNR( X ) and L = { R  R ∈ SVNR( X )} to denote the set of neutrosophic upper and lower approximation operators in X, respectively. According to Lemma 1, Remark 2 and Theorem 3, we have the following result: both ( H, ≤, ∧, ∨) and ( L, ≤, ∧, ∨) are complete lattices (it is also the onedimensional case of Reference [23]). 4. The Relationship between Complete Lattices (H,≤,∧,∨) and (L,≤,∧,∨) In this section, we will study the relationship between complete lattices ( H, ≤, ∧, ∨) and ( L, ≤, ∧, ∨). Set 3 4 A = SVNR( X )  ∀ R1 , R2 ∈ SVNR( X ), R1 ≤ R2 ⇔ R1 R2 ⇔ R1 ≤ R2 . Firstly, we will give an example to illustrate thatA is not an empty family. Example 1. Let X = { a} be a single point set, R1 and R2 are two single valued neutrosophic relations in X. (1)
If R1 ≤ R2 , then R1 R2 . In fact, if R1 ≤ R2 , then R1 ( A) R2 ( A) for each A ∈ SVNS({ a}).
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Thus, ∀ a ∈ X, TR1 ( A) ( a) ≤ TR2 ( A) ( a), IR1 ( A) ( a) ≥ IR2 ( A) ( a), and FR1 ( A) ( a) ≥ FR2 ( A) ( a). Moreover, TR1 ( a, a) ∧ TA ( a) ≤ TR2 ( a, a) ∧ TA ( a), IR1 ( a, a) ∨ I A ( a) ≥ IR2 ( a, a) ∨ I A ( a), and FR1 ( a, a) ∨ FA ( a) ≥ FR2 ( a, a) ∨ FA ( a). Considering the arbitrariness of A, in particular, take A = {< a, (1, 0, 0) >}, we have TR1 ( a, a) ≤ TR2 ( a, a), IR1 ( a, a) ≥ IR2 ( a, a) and FR1 ( a, a) ≥ FR2 ( a, a). Hence, R1 R2 . (2)
Similarly, we also can show that the following result: If R1 ≤ R2 , then R1 R2 . So, by (1), (2) and Corollary 1, we have SVNR({ a}) ∈ A, i.e., A is not an empty family. Now, we will give the relationship between complete lattices ( H, ≤, ∧, ∨) and ( L, ≤, ∧, ∨).
Proposition 1. If SVNR( X ) ∈ A, then [i∈ I Ri ] = i∈ I Ri = [i∈ I Ri ], where I is a index set, and Ri ∈ SVNR( X ) for each i ∈ I. Proof. We ﬁrst show that [i∈ I Ri ] = i∈ I Ri . Let R be a neutrosophic relation in X such that i∈ I Ri ( A) R( A) for each A ∈ SVNS( X ), then Ri ≥ R, this is equivalent to Ri R since SVNR( X ) ∈ A. Thus i∈ I Ri R. Moreover, by the construction of [i∈ I Ri ], we have i∈ I Ri [i∈ I Ri ]. On the other hand, we can show that i∈ I Ri ( A) i∈ I Ri ( A) for each A ∈ SVNS( X ). So 3 4 [i∈ I Ri ] = R ∈ SVNR( X )  ∀ A ∈ SVNS( X ), i∈ I Ri ( A) R( A) i∈ I Ri . Hence [i∈ I Ri ] = i∈ I Ri . Now, we show that i∈ I Ri = [i∈ I Ri ]. Let R be a single valued neutrosophic relation in such that i∈ I Ri ( A) R( A) for each A ∈ SVNS( X ), then Ri ≥ R, this is equivalent to Ri R since SVNR( X ) ∈ A. Thus i∈ I Ri R. Moreover, by the construction of [i∈ I Ri ]. We have i∈ I Ri [i∈ I Ri ]. On the other hand, we can show that i∈ I Ri ( A) i∈ I Ri ( A) for each A ∈ SVNS( X ). So
[i∈ I Ri ] = { R ∈ SVNR( X )  ∀ A ∈ SVNS( X ), i∈ I Ri ( A) R( A)} i∈ I Ri . Hence, [i∈ I Ri ] = i∈ I Ri . From above proved, we know that [i∈ I Ri ] = i∈ I Ri = [ j∈ J R j ]. Theorem 4. If SVNR(X ) ∈ A, then (SVNR(X ), , , ) and ( H, ≤, ∧, ∨) are complete lattice isomorphism. Proof. Deﬁne a mapping φ12 : SVNR( X ) → H as follows: ∀ R ∈ SVNR( X ), φ12 ( R) = R. Obviously, φ12 is surjective. If R1 = R2 , notice that SVNR( X ) ∈ A, we know that R1 = R2 . So φ12 is oneone. ∀{ Ri }i∈ I ⊆ SVNR( X ) and I be a index set. By Theorem 3 and Proposition 1, we have φ12 (i∈ I Ri ) = i∈ I Ri = ∨ Ri = ∨ φ12 ( Ri ), i∈ I
i∈ I
and φ12 (i∈ I Ri ) = i∈ I Ri = [i∈ I Ri ] = ∧ Ri = ∧ φ12 ( Ri ). i∈ I
i∈ I
Hence, φ12 preserves arbitrary union and arbitrary intersection. From above proved, we know that (SVNR( X ), , , ) and ( H, ≤, ∧, ∨) are complete lattice isomorphism.
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Theorem 5. If SVNR( X ) ∈ A, then (SVNR( X ), , , ) and ( L, ≤, ∧, ∨) are complete lattice isomorphism. Proof. Deﬁne a mapping φ13 : SVNR( X ) → L as follows:∀ R ∈ SVNR( X ), φ12 ( R) = R. Obviously, φ13 is surjective. If R1 = R2 , notice that SVNR( X ) ∈ A, we know that R1 = R2 . So φ13 is oneone. ∀{ Ri }i∈ I ⊆ SVNR( X ) and I be an index set. By Theorem 3 and Proposition 1, we have φ13 (i∈ I Ri ) = i∈ I Ri = ∨ Ri = ∨ φ13 ( Ri ), i∈ I
i∈ I
and φ13 (i∈ I Ri ) = i∈ I Ri = [i∈ I Ri ] = ∧ Ri = ∧ φ13 ( Ri ). i∈ I
i∈ I
Hence, φ13 preserves arbitrary union and arbitrary intersection. From the above proof, we know that (SVNR( X ), , , ) and ( L, ≤, ∧, ∨) are complete lattice isomorphism. Theorem 6. If SVNR( X ) ∈ A, then ( H, ≤, ∧, ∨) and ( L, ≤, ∧, ∨) are complete lattice isomorphism. Proof. Through Theorems 4 and 5, we immediately know that the conclusion holds. We can also prove it by the following way: Deﬁne a mapping φ23 : H → L as follows: ∀ R ∈ H, φ23 ( R) = R. Through Theorems 4 and 5, there must be one and only one R ∈ SVNR( X ) such that φ23 ( R) = R for each R ∈ L. This shows φ23 is surjective. If R1 = R2 , notice that SVNR( X ) ∈ A, we know that R1 = R2 . So φ23 is oneone. 3 4 ∀ Ri i∈ I ⊆ H and I be a index set. Through Theorem 3 and Proposition 1, we have φ23 ( ∨ Ri ) = φ23 (i∈ I Ri ) = i∈ I Ri = ∨ Ri = ∨ φ13 ( Ri ), i∈ I
i∈ I
i∈ I
and φ13 ( ∧ Ri ) = φ13 ([i∈ I Ri ]) = [i∈ I Ri ] = [i∈ I Ri ] = ∧ Ri = ∧ φ23 ( Ri ). i∈ I
i∈ I
i∈ I
Hence, φ23 preserves arbitrary union and arbitrary intersection. So, ( H, ≤, ∧, ∨) and ( L, ≤, ∧, ∨) are complete lattice isomorphism. Remark 3. Through Theorems 4–6, we can ascertain that φ12 ,φ13 and φ23 are isomorphic mappings among complete lattices. Moreover, the following diagram can commute, i.e., φ23 ◦ φ12 = φ13 (see Figure 1).
6915 ; q s t
I
+ d
I
I
/ d
Figure 1. Correspondence relationship among three complete lattices.
5. Conclusions Following the idea of multigranulation neutrosophic rough sets on a single domain as introduced by Bo et al. (2018), we gave the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In the onedimensional case, for each special SVNR( X ), we gave a
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Symmetry 2018, 10, 417
onetoone correspondence relationship between complete lattices ( H, ≤) and ( L, ≤). Unfortunately, at the moment, we haven’t solved the following problems: (1) (2)
Can the supremum and inﬁmum of the optimistic multigranulation neutrosophic rough approximation operators be given? For any set , are ( H, ≤) and ( L, ≤) isomorphic between complete lattices?
Author Contributions: H.Z. provided the idea of the paper and proved the theorems. H.Y.Z. provided helpful suggestions. Funding: This research received no external funding. Acknowledgments: The work is partly supported by the National Natural Science Foundation of China (Grant No. 61473181, 11771263 and 11671007), the Doctoral Scientiﬁc Research Foundation of Xi’an Polytechnic University (Grant No. BS1426), the Construction Funding of Xi’an Polytechnic University for Mathematics (Grant No. 107090701), and the Scientiﬁc Research Program Funded by Shaanxi Provincial Education Department (2018). Conﬂicts of Interest: The authors declare no conﬂict of interest.
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Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic; American Research Press: Rehoboth, MA, USA, 1998. Smarandache, F. Neutrosophic set—A generialization of the intuitionistics fuzzy sets. Int. J. Pure Appl. Math. 2005, 24, 287–297. Rivieccio, U. Neutrosophic logics: Prospects and problems. Fuzzy Sets Syst. 2008, 159, 1860–1868. [CrossRef] Guo, Y.; Cheng, H.D. A new neutrosophic approach to image segmentation. Pattern Recogn. 2009, 42, 587–595. [CrossRef] Guo, Y.; Sengur, A. NCM: Neutrosophic cmeans clustering algorithm. Pattern Recogn. 2015, 48, 2710–2724. [CrossRef] Salama, A.A.; Broumi, S. Roughness of neutrosophic sets. Elixir Appl. Math. 2014, 74, 26833–26837. Broumi, S.; Smarandache, F. Rough neutrosophic sets. Ital. J. Pure Appl. Math. 2014, 32, 493–502. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets, Multispace Multistruct. Google Sch. 2010, 4, 410–413. Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [CrossRef] Ye, J. Multicriteria decisionmaking method using the correlation coefﬁcient under singlevalued neutrosophic environment. Int. J. Gen. Syst. 2013, 42, 386–394. [CrossRef] Ye, J. Improved correlation coefﬁcients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making. J. Intell. Fuzzy Syst. 2014, 27, 2453–2462. Ye, S.; Ye, J. Dice similarity measure among single valued neutrosophic multisets and its applcation in medical diagnosis. Neutrosophic Sets Syst. 2014, 6, 48–53. Majumdar, P.; Samant, S.K. On similarity and entropy of neutrosophic sets. J. Intell. Fuzzy Syst. 2014, 26, 1245–1252. S¸ ahin, R.; Küçük, A. Subsethood measure for single valued neutrosophic sets. J. Intell. Fuzzy Syst. 2015, 29, 525–530. [CrossRef] Li, Z.W.; Cui, R.C. Tsimilarity of fuzzy relations and related algebraic structures. Fuzzy Sets Syst. 2015, 275, 130–143. [CrossRef] Li, Z.W.; Cui, R.C. Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators. Inf. Sci. 2015, 305, 219–233. [CrossRef] Li, Z.W.; Liu, X.F.; Zhang, G.Q.; Xie, N.X.; Wang, S.C. A multigranulation decisiontheoretic rough set method for distributed fcdecision information systems: An application in medical diagnosis. Appl. Soft Comput. 2017, 56, 233–244. [CrossRef] Yang, H.L.; Guo, Z.L.; She, Y.H.; Liao, X.W. On single valued neutrosophic relations. J. Intell. Fuzzy Syst. 2016, 30, 1045–1056. [CrossRef]
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19. 20. 21. 22. 23.
Yang, H.L.; Zhang, C.L.; Guo, Z.L.; Liu, Y.L.; Liao, X.W. A hybrid model of single valued neutrosophic sets and rough sets: Single valued neutrosophic rough set model. Soft Comput. 2017, 21, 6253–6267. [CrossRef] Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci. 1982, 11, 341–356. [CrossRef] Bao, Y.L.; Yang, H.L. On single valued neutrosophic reﬁned rough set model and its applition. J. Intell. Fuzzy Syst. 2017, 33, 1235–1248. [CrossRef] Bo, C.X.; Zhang, X.H.; Shao, S.T.; Smarandache, F. MultiGranulation Neutrosophic Rough Sets on a Single Domain and Dual Domains with Applications. Symmetry 2018, 10, 296. [CrossRef] Zhao, H.; Zhang, H.Y. A result on single valued neutrosophic reﬁned rough approximation operators. J. Intell. Fuzzy Syst. 2018, 1–8. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
387
SS symmetry Article
Fixed Points Results in Algebras of Split Quaternion and Octonion Mobeen Munir 1, *, Asim Naseem 2 , Akhtar Rasool 1 , Muhammad Shoaib Saleem 3 and Shin Min Kang 4,5, * 1 2 3 4 5
*
Division of Science and Technology, University of Education, Lahore 54000, Pakistan; [email protected] Department of Mathematics, Government College University, Lahore 54000, Pakistan; [email protected] Department of Mathematics, University of Okara, Okara 56300, Pakistan; [email protected] Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea Center for General Education, China Medical University, Taichung 40402, Taiwan Correspondence: [email protected] (M.M.); [email protected] (S.M.K.)
Received: 3 August 2018; Accepted: 14 September 2018; Published: 17 September 2018
Abstract: Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields Z p . Some characterizations of fixed points in terms of the coefficients of these polynomials are also given. Particularly, cardinalities of these fixed points have been determined depending upon the characteristics of the underlying field. Keywords: ﬁxed point; splitquaternion; quadratic polynomial; splitoctonion Subject Classiﬁcation (2010): 30C35; 05C31
1. Introduction Geometry of spacetime can be understood by the choice of convenient algebra which reveals hidden properties of the physical system. These properties are best describable by the reﬂections of symmetries of physical signals that we receive and of the algebra using in the measurement process [1–3]. Thus, we need normed division algebras with a unit element for the better understanding of these systems. For these reasons, higher dimension algebras have been an immense source of inspiration for mathematicians and physicists as their representations pave the way towards easy understanding of universal phenomenons. These algebras present nice understandings towards general rotations and describe some easy ways to consider geometric problems in mechanics and dynamical systems [4,5]. Quaternion algebra have been playing a central role in many ﬁelds of sciences such as differential geometry, human imaging, control theory, quantum physics, theory of relativity, simulation of particle motion, 3D geophones, multispectral images, signal processing including seismic velocity analysis, seismic waveform deconvolution, statistical signal processing and probability distributions (see [6–8] and references therein). It is known that rotations of 3DMinkowski spaces can be represented by the algebra of split quaternions [5]. Applications of these algebras can be traced in the study of Graphenes, Black holes, quantum gravity and Gauge theory. A classical application of split quaternion is given in [1] where Pavsic discussed spin gauge theory. Quantum gravity of 2 + 1 dimension has been described by Carlip in [2] using split quaternions. A great deal of research is in progress where authors are focused on considering matrices of quaternions and splitquaternions [9–12]. The authors in [13] gave a fast structurepreserving method to compute singular value decomposition of quaternion matrices. Split quaternions play a vital role in geometry and physical models in fourdimensional spaces as the elements of split quaternion are used to express Lorentzian rotations [14]. Particularly, Symmetry 2018, 10, 405; doi:10.3390/sym10090405
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Symmetry 2018, 10, 405
the geometric and physical applications of split quaternions require solving split quaternionic equations [15,16]. Similarly, octonion and split octonion algebras play important role in mathematical physics. In [8], authors discussed ten dimensional spacetime with help of these eight dimensional algebras. In [16], authors gave comprehensive applications of split octonions in geometry. Anastasiou developed Mtheory algebra with the help of octonions [3]. This article mainly covers ﬁnite algebras of split quaternion and split octonion over prime ﬁelds Z p . Split quaternion algebra over R was in fact introduced by James Cockle in 1849 on already established quaternions by Hamilton in 1843. Both of these algebras are actually associative, but noncommutative, nondivision ring generated by four basic elements. Like quaternion, it also forms a four dimensional real vector space equipped with a multiplicative operation. However, unlike the quaternion algebra, the split quaternion algebra contains zero divisors, nilpotent and nontrivial idempotents. For a detailed description of quaternion and its generalization (octonions), please follow [15–18]. As mathematical structures, both are algebras over the real numbers which are isomorphic to the algebra of 2 × 2 real matrices. The name split quaternion is used due to the division into positive and negative terms in the modulus function. The set (1, i,ˆ j,ˆ kˆ ) forms a basis. The product of these elements are iˆ2 = −1, jˆ2 = ˆ iˆjˆkˆ = 1. It follows from the deﬁning relations that 1 = kˆ 2 , iˆjˆ = kˆ = − jˆi,ˆ jˆkˆ = −iˆ = −kˆ j,ˆ kˆ iˆ = jˆ = −iˆk, the set (±1, ±i, ± j, ±k) is a group under split quaternion multiplication which is isomorphic to the dihedral group of a square. Following Table 1 encodes the multiplication of basis split quaternions. Table 1. Split quaternion multiplication table. . 1 iˆ jˆ kˆ
1
iˆ
jˆ
kˆ
1 iˆ jˆ kˆ
iˆ −1 −kˆ jˆ
jˆ kˆ
kˆ − jˆ −iˆ 1
1 iˆ
The split octonion is an eightdimensional algebraic structure, which is nonassociative algebra over some ﬁeld with basis 1, t´1 , t´2 , t´3 , t´4 , t´5 , t´6 and t´7 . The subtraction and addition in split octonions is computed by subtracting and adding corresponding terms and their coefﬁcients. Their multiplication is given in this table. The product of each term can be given by multiplication of the coefﬁcients and a multiplication table of the unit split octonions is given following Table 2. Table 2. Split octonions’ multiplication table. .
t´1
t´2
t´3
t´4
t´5
t´6
t´7
t´1 t´2 t´3 t´4 t´5 t´6 t´7
−1 −t´3 t´2 t´7 −´t6 t´5 −´t4
t´3 −1 −t´1 t´6 t´7 −´t4 −´t5
−t´2 t´1 −1 −t´5 t´4 t´7 −t´6
−t´7 −t´6 t´5 1 t´3 −t´2 −t´1
t´6 −t´7 −t´4 −t´3 1 t´1 −t´2
−t´5 t´4 −t´7 t´2 −t´1 1 −t´3
t´4 t´5 t´6 t´1 t´2 t´3 1
From the table, we get very useful results: t´2i = −1, ∀i = 1, ..., 3, t´2i = 1, ∀i = 4, ..., 7, t´i t´j = −t´j t´i , ∀i = j.
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Brand in [19] computed the roots of a quaternion over R. Strictly speaking, he proved mainly De Moivres theorem and then used it to ﬁnd nth roots of a quaternion. His approach paved way for ﬁnding roots of a quaternion in an efﬁcient and intelligent way. Ozdemir in [20] computed the roots of a split quaternion. In [21], authors discussed Euler’s formula and De Moivres formula for quaternions. In [15], authors gave some geometrical applications of the split quaternion. It is important to mention that these two algebras can also be constructed for Z p over prime ﬁnite ﬁelds of characteristic P. In this way, we obtain ﬁnite algebras with entirely different properties. Recently, the ring of quaternion over Z p was studied by Michael Aristidou in [22,23], where they computed the idempotents and nilpotents in H/Z p . In [18], authors computed the roots of a general quadratic polynomial in algebra of split quaternion over R. They also computed ﬁxed points of general quadratic polynomials in the same sittings. A natural question arises as to what happens with the same situations over Z p . Authors in [24] discussed splitquaternion over Z p in algebraic settings. In the present article, we ﬁrst obtain the roots of a general quadratic polynomial in the algebra of split quaternion over Z p . Some characterizations of ﬁxed points in terms of the coefﬁcients of these polynomials are also given. As a consequence, we give some computations about algebraic properties of particular classes of elements in this settings. We also give examples as well as the codes that create these examples with ease. For a computer program, we refer to Appendix A at the end of the article. We hope that our results will be helpful in understanding the communication in machine language and cryptography. Deﬁnition 1. Let x ∈ Hs , x = a0 + a1 iˆ + a2 jˆ + a3 kˆ where ai ∈ R. The conjugate of x is deﬁned as ˆ The square of pseudonorm of x is given by x¯ = a0 − a1 iˆ − a2 jˆ − a3 k. N ( x ) = x x¯ = a20 + a21 − a23 − a24 .
(1)
Deﬁnition 2. Let x = a0 + ∑7i=1 ai t´i ∈ Os /Z p . The conjugate of x is deﬁned as 7
x
= a0 + ∑ ai t´i i =1 7
= a0 + ∑ ai t´i i =1 7
= a0 − ∑ ai t´i i =1 7
= a0 + ∑ a´i t´i , i =1
where a´i = − ai where i = 1, 2, ..., 7. The square of pseudonorm of x is given by N ( x ) = xx =
3
7
i =0
i =4
∑ a2i − ∑ a2i .
2. Main Results In this section, we formulate our main results. At ﬁrst, we give these results for split quaternions and then we move towards split octonions.
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2.1. Some Fixed Points Results of Quadratic Functions in Split Quaternions over the Prime Field We ﬁrst solve a general quadratic polynomial in algebra of split quaternion. As a consequence, we ﬁnd ﬁxed points of associated functions in this algebra. Theorem 1. The quadratic equation ax2 + bx + c = 0 a, b, c ∈ Z p , where p is an odd prime and p a, has root p−b p2 − b2 x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈ Hs /Z p if and only if a0 = and a2 − a2 − a2 = ( 2 ) + c . 1
2a
2
3
4a
a
Proof. x x2
ˆ = a0 + a1 iˆ + a2 jˆ + a3 k, = ( a0 + a1 iˆ + a2 jˆ + a3 kˆ )2 ,
(2) (3)
= a20 − a21 + a22 + a23 + 2a0 a1 iˆ + 2a0 a2 jˆ + 2a0 a3 kˆ = a20 + a20 − $ x $ + 2a0 a1 iˆ + 2a0 a2 jˆ + 2a0 a3 kˆ = 2a20 − $ x $ + 2a0 a1 iˆ + 2a0 a2 jˆ + 2a0 a3 kˆ = 2a0 ( a0 + a1 iˆ + a2 jˆ + a3 kˆ ) − $ x $ = 2a0 x − $ x $. Putting x and x2 into ax2 + bx + c = 0, we have 2aa0 x − a$ x $ + bx + c
= 0,
(2aa0 + b) x − a$ x $ + c = 0, (2aa0 + b)( a0 + a1 iˆ + a2 jˆ + a3 kˆ ) − a( a20 + a21 − a22 − a23 ) + c = 0, (2aa0 + b) a0 + (2aa0 + b)( a1 iˆ + a2 jˆ + a3 kˆ ) − a( a20 + a21 − a22 − a23 ) + c = 0. Comparing vector terms in the above equation, we get 2aa0 + b a0
= 0, −b p−b = = . 2a 2a
(4) (5)
Comparing constant terms, we get
(2aa0 + b) a0 − a( a20 + a21 − a22 − a23 ) + c = 0,
(6)
(2aa0 + b) a0 − aa20 + c = a( a21 − a22 − a23 ), aa20 + ba0 + c
( aa0 + b) a0 + c p−b p−b ( a( ) + b) +c 2a 2a 2 2 c p −b + a 4a2
= a( a21 − a22 − a23 ), =
a( a21
−
a22
=
a( a21
−
a22
(8)
−
a23 ),
(9)
−
a23 ),
(10)
= a21 − a22 − a23 .
On the basis of the above results 2.1, we arrive at a new result given as
391
(7)
(11)
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Theorem 2. The ﬁxed point of function f ( x ) = x2 + (b + 1) x + c where a, b, c ∈ Z p , p is an odd prime and p2 − b2 p−b and a2 − a2 − a2 = ( 2 ) + c . p a is x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈ Hs /Z p if and only if a0 = 2a
1
2
3
4a
a
Proof. It is enough to give a new relation f ( x ) = g( x ) + x, where g( x ) = + bx + c. Then, existence of ﬁxed points for f ( x ) is equivalent to the solutions of g( x ). Then, the required result is immediate from the above theorem. x2
Theorem 3. Let p be an odd prime, p a, if x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈ Hs /Z p is a root of quadratic equation x2 + bx + c = 0, where a, b, c ∈ Z p . Then, conjugate of x i.e., x¯ = a0 − a1 iˆ − a2 jˆ − a3 kˆ ∈ Hs /Z p is also the root of quadratic equation x2 + bx + c = 0. Proof. The proof follows simply by using condition of Theorem 1 applied on the conjugate of x. Theorem 4. Let p be an odd prime, p a, if x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈ Hs /Z p be the ﬁxed point of function f ( x ) = x2 + (b + 1) x + c, where a, b, c ∈ Z p . Then, the conjugate of x i.e., x¯ = a0 − a1 iˆ − a2 jˆ − a3 kˆ ∈ Hs /Z p also be the ﬁx point of function f ( x ) = x2 + (b + 1) x + c. Proof. Again, it is enough to use relation f (x) = g(x) + x where g(x) = x2 + bx + c. Then, the existence of fixed points for f (x) is equivalent to the solutions of g(x). Then, the required result is immediate from the above theorem. The following two theorems are new results about the number of ﬁxed points of f ( x ) = x2 + (b + 1) x + c. ( Theorem 5.  Fix ( f ) =
p2 , b = 0, c = 0, p2 + p + 2, c = 0, b = 0.
Proof. We split the proof in cases.
∼ M2 (Z p ),where p is prime. It is easy to see Case 1: For c = 0 and b = 0, we obtain two Hs /Z p = that Hs /Z p and M2 (Z p ) are isomorphic as algebras, the map ϕ : Hs /Z p *−→ M2 (Z p ) is deﬁned as 0p−1 0p−1 p−10 ∼ ϕ( a0 + a1 iˆ + a2 jˆ + a3 kˆ ) = a0 (10 01) + a1 ( 10 ) + a2 ( p−10) + a3 ( 01 ). As Hs /Z p = M2 (Z p ), so we ﬁnd the number of nilpotent elements in M2 (Z p ). It is wellknown by Fine and Herstein that the probability that n × n matrix over a Galois ﬁeld having pα elements have pα.n nilpotent elements. As in our case, α = 1 and n = 2, thus the probability that the 2 × 2 matrix over Z p has p−2 nilpotent elements: nil (M2 (Z p )) (M2 (Z p )) nil (M2 (Z p )) p4
=
p −2 ,
(12)
=
p −2 ,
(13)
2
(14)
nil (M2 (Z p )) =
p .
Case 2: For c = 0 and b = 0, we obtain as many points as there are matrices M2 (Z p ) because of the above isomorphism, and, using the argument given in 2, we arrive at the result. ⎧ 2 ⎪ ⎨ p − p, p ≡ 1(mod3), Theorem 6. Let b = 0 and c = 0. Then,  Fix ( f ) = p2 + p, p ≡ 2(mod3), ⎪ ⎩ 3, p = 3. Proof. Case 1: For p = 3, there is nothing to prove. Case 2: For p ≡ 1(mod3), we have two further cases:
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case I: If p ≡ 3(mod4), x2 + y2 = z has a unique solution for z = 0. x2 + y2 = z has ( p + 1) options for z = 0, thus ( p + 1)( p − 1) options in all. Thus, we get that x2 + y2 = z has total number of solutions ( p + 1)( p − 1) + 1 = p2 − 1 + 1 = p2 . Now, when z = 0, we get no solution for a1 :
=
p−1 )( p + 1) 2 p + 1 + p2 − 1
(16)
=
p2 + p.
(17)
= 1( p + 1) + 2(
(15)
case II: If p ≡ 1(mod4), x2 + y2 = z has (2p − 1) solutions for z = 0. x2 + y2 = z has ( p − 1) options for z = 0, thus ( p − 1)( p − 1) options in all. Thus, we get that x2 + y2 = z has total number of solutions ( p − 1)( p − 1) + (2p − 1) = p2 − p − p + 1 + 2p − 1 = p2 . Now, when z = 0, we get two solutions for a1 :
=
p−3 )( p − 1) + 1( p − 1) 2 4p − 2 + p2 − p − 3p + 3 + p − 1
=
p + p.
= 2(2p − 1) + 2( 2
(18) (19) (20)
Case 3: For p ≡ 2(mod3), we have two further cases: case I: If p ≡ 3(mod4) x2 + y2 = z has a unique solution for z = 0. x2 + y2 = z has ( p + 1) options for z = 0. So ( p + 1)( p − 1) options in all. Thus we get, x2 + y2 = z has total number of solutions ( p + 1)( p − 1) + 1 = p2 − 1 + 1 = p2 Now, when z = 0, we get no solution for a1 :
=
p−3 )( p + 1) + 1( p + 1) 2 2 2 + p + p − 3p − 3 + p + 1
=
p − p.
= 1(2) + 2( 2
(21) (22) (23)
case II: If p ≡ 1(mod4), x2 + y2 = z has (2p − 1) solutions for z = 0. x2 + y2 = z has ( p − 1) options for z = 0. So ( p − 1)( p − 1) options in all. Thus we get, x2 + y2 = z has total number of solutions ( p − 1)( p − 1) + (2p − 1) = p2 − p − p + 1 + 2p − 1 = p2 . Now, when z = 0, we get two solutions for a1 .
=
p−1 )( p − 1) 2 2 p−1+ p − p− p+1
=
p − p.
= 1( p − 1) + 2( 2
(24) (25) (26)
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2.2. Some Algebraic Consequences about Hs /Z p We can understand the algebraic structure of Hs /Z p with ease. The following results are simple facts obtained from the previous section. Corollary 1. Let p be an odd prime, an element x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈ Hs /Z p is idempotent ⇔ a0 =
p +1 2
and a21 − a22 − a23 =
(27)
p2 −1 4 .
Proof. Taking a = 1, b = p − 1 and c = p in the above theorem, we have x 2 + ( p − 1) x + p
= 0,
x −x
= 0,
2
x2
=
x
has root ˆ x = a0 + a1 iˆ + a2 jˆ + a3 k,
where a0
= =
p−b 2a p+1 , 2
and a21 − a22 − a23
=
p2 − b2 c p2 − (−1)2 0 + = + a 1 4a2 4(1)2
=
p2 − 1 . 4
In other words, we can say x is idempotent. We also present similar results but without proof as they can be derived similarly. Corollary 2. Let p be an odd prime an element and x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈ Hs /Z p is idempotent if and only if a0 =
p +1 2
(28)
and $ x $ = 0.
Corollary 3. Let p be an odd prime and x ∈ Hs /Z p . If x is an idempotent, then $ x $ = 0. Corollary 4. Let p be an odd prime. If x ∈ Hs /Z p is idempotent, then x¯ is also an idempotent. Corollary 5. Let p be an odd prime. If x ∈ Hs /Z p and x is of the form x = a0 . If x is idempotent, then it is either 0 or 1.
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Corollary 6. Let p be an odd prime and x ∈ Hs /Z p of the form ˆ x = a0 + a1 iˆ + a2 jˆ + a3 k,
(29)
where at least one ai = 0. Then, x is not an idempotent. Corollary 7. Let p be an odd prime, and the quadratic equation x2 = 0 has root x = a0 + a1 iˆ + a2 jˆ + a3 kˆ ∈
Hs /Z p , where a0 =
p 2
and a21 − a22 − a23 =
p2 4 .
Proof. Taking a = 1, b = 0 and c = 0 in the above theorem, we have that x2 + ( p) x + o x
2
= 0, = 0
has root ˆ x = a0 + a1 iˆ + a2 jˆ + a3 k, where
=
a0
=
p−b p−0 = 2a 2 p , 2
and a21 − a22 − a23
=
p2 − b2 c p2 − (0)2 0 + = + 2 a 1 4a 4(1)2
a21 − a22 − a23
=
p2 . 4
In other words, we can say x is nilpotent. 2.3. Some Fixed Points Results of Quadratic Functions in Split Octonions over the Prime Field Theorem 7. The quadratic equation ax2 + bx + c = 0 where a, b, c ∈ Z p , p is an odd prime and p a has root p−b p2 − b2 x = a0 + ∑7i=1 ai t´i ∈ Os /Z p if and only if a0 = and ∑3i=1 a2 − ∑7i=4 a2 = ( 2 ) + c . 2a
i
Proof. ax2 + bx + c = 0.
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i
4a
a
Symmetry 2018, 10, 405
Take x = a0 + ∑7i=1 ai t´i , we have x2
7
= ( a0 + ∑ ai t´i )2 i =1
7
7
= ( a0 )2 + ( ∑ ai t´i )2 + 2a0 ∑ ai t´i i =1 7
i =1
7
7
i =1
i =1 7
7
= ( a0 )2 − ∑ a2i + 2a0 ∑ ai t´i where( ∑ ai t´i )2 = − ∑ a2i i =1
= ( a0 ) + 2
a20
i =1
7
−  x  + 2a0 ∑ ai t´i where x  = a0 + ∑ a2i i =1
i =1
7
= 2( a0 )2 −  x  + 2a0 ∑ ai t´i i =1
7
= 2( a0 )2 + 2a0 ∑ ai t´i −  x , i =1
= 2a0 x −  x . Putting it in the above equation, we get a(2a0 x −  x ) + bx + c
= 0,
(30)
2aa0 x − a x  + bx + c
= 0,
(31)
(2aa0 + b) x − a x  + c = 0.
(32)
Here, x = a0 + ∑7i=1 ai t´i and  x  = a20 + ∑3i=1 a2i − ∑7i=3 a2i , we have 7
3
7
(2aa0 + b)( a0 + ∑ ai t´i ) − a[ a20 + ∑ a2i − ∑ a2i ] + c = o, i =1 7
(2aa0 + b) a0 + (2aa0 + b) ∑ ai t´i − i =1
aa20
i =1 3
−a∑
i =1
a2i
i =3 7
+ a ∑ a2i + c = o, i =3
Comparing vector terms on both sides, we have
(2aa0 + b) ai
= 0,
2aa0 + b
= 0, −b = , 2a p−b . = 2a
a0 a0
Comparing constant terms on both sides, we have 3
7
i =1
i =4
(2aa0 + b) a0 − aa20 − a ∑ a2i + a ∑ a2i + c = o, 2aa20 + ba0 − aa20 + c a0 ( aa0 + b) + c
396
3
7
= a ∑ a2i − a ∑ a2i , i =1 3
= a[ ∑
i =1
a2i
i =4 7
− ∑ a2i ], i =4
Symmetry 2018, 10, 405
where a0 =
p−b 2a .
(
p−b p−b )( a( ) + b) + c 2a 2a
3
7
= a[ ∑ a2i − ∑ a2i ], i =1 3
p−b p+b ( )( )+c 2a 2
= a[ ∑
i =1 3
a2i
i =4 7
− ∑ a2i ], i =4 7
p2 − b2 ( )+c 4a
= a[ ∑ a2i − ∑ a2i ],
p2 − b2 c ( )+ a 4a2
=
i =1 3 a2i i =1
∑
i =4 7 − a2i . i =4
∑
Theorem 8. The ﬁxed points of function f ( x ) = ax2 + (b + 1) x + c are x = a0 + ∑7i=1 ai t´i ∈ Os /Z p , where a0 =
p−b 2a
and ∑3i=1 a2i − ∑7i=4 a2i = (
p2 − b2 ) + ac . 4a2
Proof. It is enough to use relation f ( x ) = g( x ) + x where g( x ) = ax2 + bx + c. Then, the existence of ﬁxed points for f ( x ) is equivalent to the solutions of g( x ). Then, the required result is immediate from the above theorem. Corollary 8. The ﬁxed point of function f ( x ) = x2 + x are x = a0 + ∑7i=1 ai t´i ∈ Os /Z p where a0 = ∑3i=1 a2i
− ∑7i=4 a2i
=
p2 4 .
p 2
and
Proof. It is obvious from the above theorem, only by taking a = 1, b = 0 and c = 0. Theorem 9. Let p be an odd prime. If x = a0 + ∑7i=1 ai t´i ∈ Os /Z p is the root of the quadratic equation ax2 + bx + c = 0 a, b, c ∈ Z p , then x = a0 + ∑7i=1 a´i t´i ∈ Os /Z p is also the root of the quadratic equation ax2 + bx + c = 0 a, b, c ∈ Z p . Proof. x
7
7
7
i =1 7
i =1
i =1
= a0 + ∑ ai t´i = a0 + ∑ a´i ti = a0 − ∑ ai t´i
(33)
= a0 + ∑ a´i t´i ,
(34)
i =1
where a´i = − ai where i = 1, 2, ..., 7 as a0 =
p−b 2a
and 3
7
i =1
i =4
∑ a´i 2 − ∑ a´i 2
= = =
3
7
i =1 3
i =4
∑ (−ai )2 − ∑ (−ai )2 7
∑ ( a i )2 − ∑ ( a i )2
i =1 p2 − b2
4a2
(36)
i =4
c + . a
It implies that x is the root of the quadratic equation ax2 + bx + c = 0 a, b, c ∈ Z p .
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Theorem 10. If the function f ( x ) = ax2 + (b + 1) x + c has ﬁxed point x = a0 + ∑7i=1 ai t´i ∈ Os /Z p , then x = a0 + ∑7i=1 a´i t´i ∈ Os /Z p also is the ﬁxed point of function f ( x ) = ax2 + (b + 1) x + c. Proof. It is enough to use relation f ( x ) = g( x ) + x, where g( x ) = ax2 + bx + c. Then, the existence of ﬁxed points for f ( x ) is equivalent to the solutions of g( x ). Then, the required result is immediate from the above theorem. 3. Some Algebraic Consequences about Os /Z p Proposition 1. Let p be an odd prime and an element 7
x = a0 + ∑ ai t´i ∈ Os /Z p i =1
is idempotent ⇔ a0 =
p +1 2
and 3
7
i =1
i =4
∑ a2i − ∑ a2i =
p2 − 1 . 4
Proof. Taking a = 1, b = p − 1 and c = p in the above theorem, we have x 2 + ( p − 1) x + p
= 0,
(38)
x2 − x
= 0,
(39)
=
(40)
x
2
x
has root 7
x = a0 + ∑ ai t´i ,
(41)
i =1
where a0
= =
p−b p− p+1 = 2a 2 1 , 2
(42) (43)
and 3
7
i =1
i =4
∑ a2i − ∑ a2i
=
p2 − b2 c p2 − (−1)2 0 + = + 2 a 1 4a 4(1)2
(44)
=
p2 − 1 . 4
(45)
In other words, we can say that x is idempotent. Proposition 2. Let p be an odd prime and element 7
x = a0 + ∑ ai t´i ∈ Os /Z p i =1
is idempotent if and only if a0 =
p +1 2
and $ x $ = 0.
Proposition 3. Let p be an odd prime and x ∈ Os /Z p . If x is an idempotent, then $ x $ = 0.
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Proposition 4. Let p be an odd prime. If x ∈ Os /Z p is idempotent, then x¯ is also an idempotent. Proposition 5. Let p be an odd prime. If x = a0 ∈ Os /Z p is idempotent, then it is either 0 or 1. Proposition 6. Let p be an odd prime and x ∈ Os /Z p be of the form x=
7
∑ ai t´i ,
(47)
i =1
where at least one ai = 0. Then, x is not an idempotent. Proposition 7. Let p be an odd prime and the quadratic equation x2 = 0 has root x = a0 + ∑7i=1 ai t´i ∈
Os /Z p , where a0 =
p 2
and ∑3i=1 a2i − ∑7i=4 a2i =
p2 4 .
Proof. Taking a = 1, b = 0 and c = 0 in the above theorem, we have x2 + ( p) x + o
= 0,
(48)
x2
= 0
(49)
has root 7
x = a0 + ∑ ai t´i ,
(50)
i =1
where a0
= =
p−b p−0 = 2a 2 p 2
(51) (52)
and 3
7
i =1
i =4
∑ a2i − ∑ a2i
=
p2 − b2 c p2 − (0)2 0 + = + 2 a 1 4a 4(1)2
(53)
=
p2 . 4
(54)
In other words, we can say that x is nilpotent. Using results of the previous section and programs mentioned in the Appendix A, we can give many examples. 4. Examples In this section, we add examples relating to the previous section. These results are generated by the codes given in Appendix A. These along with other examples can be created using codes, and results can be applied to crypto systems and communication channel systems.
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Example 1. We ﬁnd all solutions of x2 − x = 0 over Hs /Z7 . As above, we see that, if x ∈ Hs /Z7 , then 7−(−1) a0 = = 4 and following are the values of a1 , a2 and a3 , respectively, satisfying the equation − a21 + a22 + 2 a23 = −5 or a21 − a22 − a23 = 5.
(0, 1, 1) (1, 1, 4) (2, 3, 2) (3, 0, 2) (4, 0, 2) (5, 0, 6) (5, 5, 4)
(0, 1, 6) (1, 3, 1) (2, 3, 3) (3, 0, 5) (4, 0, 5) (5, 2, 3) (5, 6, 0)
(0, 3, 0) (1, 3, 6) (2, 3, 5) (3, 2, 0) (4, 2, 0) (5, 2, 4) (6, 1, 4)
(0, 4, 0) (1, 4, 1) (2, 4, 2) (3, 3, 3) (4, 3, 3) (5, 3, 2) (6, 3, 6)
(0, 1, 1) (1, 4, 6) (2, 4, 3) (3, 3, 4) (4, 3, 4) (5, 3, 5) (6, 4, 1)
(0, 6, 1) (1, 6, 4) (2, 4, 5) (3, 4, 3) (4, 4, 3) (5, 4, 2) (6, 4, 6)
(0, 6, 6) (2, 2, 3) (2, 5, 3) (3, 4, 4) (4, 4, 4) (5, 4, 5) (6, 6, 3)
(1, 1, 3) (2, 2, 4) (2, 5, 4) (3, 5, 0) . (4, 5, 0) (5, 5, 3) (6, 6, 4)
Example 2. We compute all solutions of 2x2 + x = 0 over Hs /Z5 . As above, we see that, if x ∈ Hs /Z5 , then a0 = 1 and following are the values of a1 , a2 and a3 , respectively, satisfying the equation − a21 + a22 + a23 = −4 or a21 − a22 − a23 = 4.
(0, 0, 1) (0, 0, 4) (0, 1, 0) (0, 4, 0) (1, 1, 1) (1, 1, 4)
(1, 4, 1) (1, 4, 4) (2, 0, 0) (2, 1, 2) (2, 1, 3) (2, 2, 1)
(2, 2, 4) (2, 3, 1) (2, 3, 4) (2, 4, 2) (2, 4, 3) (3, 0, 0)
(3, 1, 2) (3, 1, 3) (3, 2, 1) (3, 2, 4) (3, 3, 1) (3, 3, 4)
(3, 4, 2) (3, 4, 3) (4, 1, 1) . (4, 1, 4) (4, 4, 1) (4, 4, 4)
Example 3. We compute all solutions of x2 + x + 1 = 0 over Hs /Z7 . As above, we see that, if x ∈ Hs /Z7 , then a0 = 3 and following are the values of a1 , a2 and a3 , respectively satisfying the equation − a21 + a22 + a23 = −6 or a21 − a22 − a23 = 6.
(0, 0, 1) (0, 0, 6) (0, 1, 0) (0, 2, 2) (0, 2, 5) (0, 5, 2) (0, 5, 5) (0, 6, 0)
(1, 0, 3) (1, 0, 4) (1, 1, 6) (1, 3, 0) (1, 4, 0) (1, 6, 1) (1, 1, 1) (1, 6, 6)
(6, 0, 3) (6, 0, 4) (6, 1, 6) (6, 3, 0) (6, 4, 0) (6, 6, 1) (6, 1, 1) (6, 6, 6)
(2, 1, 2) (2, 1, 5) (2, 2, 1) (2, 2, 6) (2, 5, 1) (2, 5, 6) (2, 6, 2) (2, 6, 5)
(5, 1, 2) (5, 1, 5) (5, 2, 1) (5, 2, 6) (5, 5, 1) (5, 5, 6) (5, 6, 2) (5, 6, 5)
(3, 1, 3) (3, 1, 4) (3, 3, 1) (3, 3, 6) (3, 4, 1) (3, 4, 6) (3, 6, 3) (3, 6, 4)
(4, 1, 3) (4, 1, 4) (4, 3, 1) (4, 3, 6) . (4, 4, 1) (4, 4, 6) (4, 6, 2) (4, 6, 5)
Example 4. We compute all solutions of x2 = 0 over Hs /Z5 . As above, we see that, if x ∈ Hs /Z5 , then a0 = 0 and following are the values of a1 , a2 and a3 , respectively, satisfying the equation − a21 + a22 + a23 = 0 or a21 − a22 − a23 = 0.
(0, 0, 0) (0, 1, 2) (0, 1, 3) (0, 2, 1) (0, 2, 4)
(0, 3, 4) (0, 4, 2) (0, 4, 3) (1, 0, 1) (1, 0, 4)
(1, 4, 0) (4, 0, 1) (4, 0, 4) (4, 1, 0) (4, 4, 0)
(2, 0, 3) (2, 2, 0) (2, 3, 0) (3, 0, 2) (3, 0, 3)
(3, 3, 0) (0, 3, 1) (1, 1, 0) . (2, 0, 2) (3, 2, 0)
Example 5. We compute all solutions of x2 − x = 0 over Os /Z3 (idempotents in the split octonion algebra). 3−(−1) As above we see that x = a0 + ∑7i=1 ai t´i ∈ Os /Z3 where a0 = = 2 and following is the values of a1 , 2 a2 , a3 , a4 , a5 , a6 and a7 respectively satisfying the equation ∑3i=1 a2i − ∑7i=4 a2i = ( We do so by putting values for p = 3, a = 1, b = −1, c = 0 in above given code.
400
p2 − b2 ) + ac 4a2
=2.
Symmetry 2018, 10, 405
(2, 1, 0, 1, 1, 0, 2); (2, 1, 0, 1, 1, 1, 0); (2, 1, 0, 1, 1, 2, 0); (2, 1, 0, 1, 2, 0, 1); (2, 1, 0, 1, 2, 0, 2); (2, 1, 0, 1, 2, 1, 0); (2, 1, 0, 1, 2, 2, 0); (2, 1, 0, 2, 0, 1, 1); (2, 1, 0, 2, 0, 1, 2); (2, 1, 0, 2, 0, 2, 1); (2, 1, 0, 2, 0, 2, 2); (2, 1, 0, 2, 1, 0, 1); (2, 1, 0, 2, 1, 0, 2); (2, 1, 0, 2, 1, 1, 0); (2, 1, 0, 2, 1, 2, 0); (2, 1, 0, 2, 2, 0, 1); (2, 1, 0, 2, 2, 0, 2); (2, 1, 0, 2, 2, 1, 0); (2, 1, 0, 2, 2, 2, 0); (2, 1, 1, 0, 0, 0, 1); (2, 1, 1, 0, 0, 0, 2); (2, 1, 1, 0, 0, 1, 0); (2, 1, 1, 0, 0, 2, 0); (2, 1, 1, 0, 1, 0, 0); (2, 1, 1, 0, 2, 0, 0); (2, 1, 1, 1, 0, 0, 0); (2, 1, 1, 1, 1, 1, 1); (2, 1, 1, 1, 1, 1, 2); (2, 1, 1, 1, 1, 2, 1); (2, 1, 1, 1, 1, 1, 1); (2, 1, 1, 1, 1, 1, 2); (2, 1, 1, 1, 1, 2, 1); (2, 1, 1, 1, 1, 2, 2); (2, 1, 1, 1, 2, 1, 1); (2, 1, 1, 1, 2, 1, 2); (2, 1, 1, 1, 2, 2, 1); (2, 1, 1, 1, 2, 2, 2); (2, 1, 1, 2, 0, 0, 0); (2, 1, 1, 2, 1, 1, 1); (2, 1, 1, 2, 1, 1, 2); (2, 1, 1, 2, 1, 2, 1); (2, 1, 1, 2, 1, 2, 2); (2, 1, 1, 2, 2, 1, 1); (2, 1, 1, 2, 2, 1, 2); (2, 1, 1, 2, 2, 2, 1); (2, 1, 1, 2, 2, 2, 2); (2, 1, 2, 0, 0, 0, 1); (2, 1, 2, 0, 0, 0, 2); (2, 1, 2, 0, 0, 1, 0); (2, 1, 2, 0, 0, 2, 0); (2, 1, 2, 0, 1, 0, 0); (2, 1, 2, 0, 2, 0, 0); (2, 1, 2, 1, 0, 0, 0); (2, 1, 2, 1, 1, 1, 1); (2, 1, 2, 1, 1, 1, 2); (2, 1, 2, 1, 1, 2, 1); (2, 1, 2, 1, 1, 2, 2); (2, 1, 2, 1, 2, 1, 1); (2, 1, 2, 1, 2, 1, 2); (2, 1, 2, 1, 2, 2, 1); (2, 1, 2, 1, 2, 2, 2); (2, 1, 2, 2, 0, 0, 0); (2, 1, 2, 2, 1, 1, 1); (2, 1, 2, 2, 1, 1, 2); (2, 1, 2, 2, 1, 2, 1); (2, 1, 2, 2, 1, 2, 2); (2, 1, 2, 2, 2, 1, 1); (2, 1, 2, 2, 2, 1, 2); (2, 1, 2, 2, 2, 2, 1); (2, 1, 2, 2, 2, 2, 2); (2, 2, 0, 0, 0, 0, 0); (2, 2, 0, 0, 1, 1, 1); (2, 2, 0, 0, 1, 1, 2); (2, 2, 0, 0, 1, 2, 1); (2, 2, 0, 0, 1, 2, 2); (2, 2, 0, 0, 1, 2, 2); (2, 2, 0, 0, 2, 1, 1); (2, 2, 0, 0, 2, 1, 2); (2, 2, 0, 0, 2, 2, 1); (2, 2, 0, 0, 2, 2, 2); (2, 2, 0, 1, 0, 1, 1); (2, 2, 0, 1, 0, 1, 2); (2, 2, 0, 1, 0, 2, 1); (2, 2, 0, 1, 0, 2, 2); (2, 2, 0, 1, 1, 0, 1); (2, 2, 0, 1, 1, 0, 2); (2, 2, 0, 1, 1, 1, 0); (2, 2, 0, 1, 1, 2, 0); (2, 2, 0, 1, 2, 0, 1); (2, 2, 0, 1, 2, 0, 2); (2, 2, 0, 1, 2, 1, 0); (2, 2, 0, 1, 2, 2, 0); (2, 2, 0, 2, 0, 1, 1); (2, 2, 0, 2, 0, 1, 1); (2, 2, 0, 2, 0, 1, 2); (2, 2, 0, 2, 0, 2, 1); (2, 2, 0, 2, 0, 2, 2); (2, 2, 0, 2, 1, 0, 1); (2, 2, 0, 2, 1, 0, 2); (2, 2, 0, 2, 1, 1, 0); (2, 2, 0, 2, 1, 2, 0); (2, 2, 0, 2, 2, 0, 1); (2, 2, 0, 2, 2, 0, 2); (2, 2, 0, 2, 2, 1, 0); (2, 2, 0, 2, 2, 2, 0); (2, 2, 1, 0, 0, 0, 1); (2, 2, 1, 0, 0, 0, 2); (2, 2, 1, 0, 0, 1, 0); (2, 2, 1, 0, 0, 2, 0); (2, 2, 1, 0, 1, 0, 0); (2, 2, 1, 0, 2, 0, 0); (2, 2, 1, 1, 0, 0, 0); (2, 2, 1, 1, 1, 1, 1); (2, 2, 1, 1, 1, 1, 1); (2, 2, 1, 1, 1, 1, 2); (2, 2, 1, 1, 1, 2, 1); (2, 2, 1, 1, 1, 2, 2); (2, 2, 1, 1, 2, 1, 1); (2, 2, 1, 1, 2, 1, 2); (2, 2, 1, 1, 2, 2, 1); (2, 2, 1, 1, 2, 2, 2); (2, 2, 1, 2, 0, 0, 0); (2, 2, 1, 2, 1, 1, 1); (2, 2, 1, 2, 1, 1, 2); (2, 2, 1, 2, 1, 2, 1); (2, 2, 1, 2, 1, 2, 2); (2, 2, 1, 2, 2, 1, 1); (2, 2, 1, 2, 2, 1, 2); (2, 2, 1, 2, 2, 2, 1); (2, 2, 1, 2, 2, 2, 2); (2, 2, 2, 0, 0, 0, 1); (2, 2, 2, 0, 0, 0, 2); (2, 2, 2, 0, 0, 1, 0); (2, 2, 2, 0, 0, 2, 0); (2, 2, 2, 0, 1, 0, 0); (2, 2, 2, 0, 2, 0, 0); (2, 2, 2, 1, 0, 0, 0); (2, 2, 2, 1, 1, 1, 1); (2, 2, 2, 1, 1, 1, 2); (2, 2, 2, 1, 1, 2, 1); (2, 2, 2, 1, 1, 2, 2); (2, 2, 2, 1, 2, 1, 2); (2, 2, 2, 1, 2, 2, 1); (2, 2, 2, 1, 2, 2, 2); (2, 2, 2, 2, 0, 0, 0); (2, 2, 2, 2, 1, 1, 1); (2, 2, 2, 2, 1, 1, 2); (2, 2, 2, 2, 1, 2, 1); (2, 2, 2, 2, 1, 2, 2); (2, 2, 2, 2, 2, 1, 1); (2, 2, 2, 2, 2, 1, 2); (2, 2, 2, 2, 2, 2, 1); (2, 2, 2, 2, 2, 2, 2).
401
Symmetry 2018, 10, 405
5. Conclusions and Further Directions In this article, we produced some general results about ﬁxed points of a general quadratic polynomial in algebras of split quaternion and octonion over Z p . We not only characterized these points in terms of the coefﬁcients of these polynomials but also gave the cardinality of these points and also the programs that produced ﬁxed points. We arrived at the following new results for a general quadratic function. ( Theorem 11.  Fix ( f ) =
p2 , b = 0, c = 0, p2 + p + 2, c = 0, b = 0.
⎧ 2 ⎪ ⎨ p − p, p ≡ 1(mod3); Theorem 12. Let b = 0 and c = 0. Then,  Fix ( f ) = p2 + p, p ≡ 2(mod3); ⎪ ⎩ 3, p = 3. We also give the following two new results for the ﬁxed points of a general quadratic quaternionic equation without proofs. Proofs are left as an open problem. ( Theorem 13.  Fix ( f ) =
p6 , p6 + p3 ,
b = 0, c = 0; c = 0, b = 0.
⎧ 6 3 ⎪ ⎨ p +p , Theorem 14. Let b = 0 and c = 0. Then,  Fix ( f ) = p6 − p3 , ⎪ ⎩ p6 ,
p ≡ 1(mod3); p ≡ 2(mod3); p = 3.
We like to remark that new results can be obtained for a general cubic polynomials in these algebras. 6. Data Availability Statement No such data has been used to prove these results. Author Contributions: M.M. conceived the idea and drafted the manuscript. Computations have been done by A.R., A.N. and M.S.S., S.M.K. edited and made ﬁnal corrections. All authors read and approved the ﬁnal manuscript. Funding: This research received no external funding. Acknowledgments: The authors are thankful to the reviewers for their valuable comments and suggestions. The authors are thankful to the University of Education Lahore for providing us a platform to present this article at ICE 2018. Conﬂicts of Interest: The authors declare that they have no competing interests.
Appendix A. Computer Codes Here, we put together some programs to compute ﬁxed points and roots easily. Appendix A.1. Program for Finding Solutions of the Quadratic Equation in Hs /Z p Following codes, count and print the number of solutions of quadratic equation ax2 + bx + c = 0 p−b in Hs /Z p . These codes print the string a1 , a2 , a3 with the understanding that the coefﬁcient a0 = 2a is ﬁxed in Hs /Z p and satisfying the relation a21 − a22 − a23 = ( for Hs /Z p .
p2 − b2 ) + ac 4a2
or − a21 + a22 + a23 = (
− p2 + b2 ) − ac 4a2
CODE: This code will give solutions of the quadratic equation only by putting values for p, a, b, c, where p is an odd prime and a, b, cεZ p . #include 402
Symmetry 2018, 10, 405
#include using namespace std; main() { int a1, a2, a3, p, n, a, b, c, count; count=0; coutp; couta; coutb; coutc; n=((p*pb*b)/(4*a*a))+(c/a); while(n