Discrete Mathematics and Symmetry
By Angel Garrido
Contents
About the Special Issue Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Preface to ”Discrete Mathematics and Symmetry” . . . . . . . . . . . . . . . . . . . . . . . . . . xi
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’c-Raˇsovi’c and Irina Cristea
Weak Embeddable Hypernear-Rings
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 XiaoyingWu
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, Jia-Bao Liu,
Mohammad Reza Farahani
Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of Generalized
Bridge Molecular Graphs
Reprinted from: Symmetry 2018, 10, 751, doi:10.3390/sym10120751 . . . . . . . . . . . . . . . . . 151
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
Jia-Bao 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
Zhan-Ao Xue, Dan-Jie Han, Min-Jie Lv and Min Zhang
Novel Three-Way Decisions Models with Multi-Granulation 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
Q-Filters of Quantum B-Algebras 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 Geyer-Schulz
Invariant Graph Partition Comparison Measures
Reprinted from: Symmetry 2018, 10, 504, doi:10.3390/sym10100504 . . . . . . . . . . . . . . . . . 266
Yanlan Mei, Yingying Liang and Yan Tu
A Multi-Granularity 2-Tuple 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 Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal
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
Zhan-ao Xue, Min-jie Lv, Dan-jie Han and Xian-wei Xin
Multi-Granulation 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
Hu Zhao and Hong-Ying 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, ZhongkaiWang and JiaxiWang
Optimizing the High-Level 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 Jae-Hyouk Lee
Binary Icosahedral Group and 600-Cell
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