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El. knyga: Advanced Topics in Fuzzy Graph Theory

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This book builds on two recently published books by the same authors on fuzzy graph theory. Continuing in their tradition, it provides readers with an extensive set of tools for applying fuzzy mathematics and graph theory to social problems such as human trafficking and illegal immigration. Further, it especially focuses on advanced concepts such as connectivity and Wiener indices in fuzzy graphs, distance, operations on fuzzy graphs involving t-norms, and the application of dialectic synthesis in fuzzy graph theory. Each chapter also discusses a number of key, representative applications. Given its approach, the book provides readers with an authoritative, self-contained guide to – and at the same time an inspiring read on – the theory and modern applications of fuzzy graphs. For newcomers, the book also includes a brief introduction to fuzzy sets, fuzzy relations and fuzzy graphs.

1 Preliminaries
1(14)
1.1 Fuzzy Sets and Relations
1(3)
1.2 Fuzzy Graphs
4(11)
2 Connectivity in Fuzzy Graphs
15(40)
2.1 Vertex Connectivity of Fuzzy Graphs
15(8)
2.2 Fuzzy Vertex Connectivity of Complement of a Fuzzy Graph
23(3)
2.3 Average Fuzzy Vertex Connectivity
26(9)
2.4 Uniformly t-Connected Fuzzy Graphs
35(2)
2.5 Application to Human Trafficking
37(5)
2.6 Critical Blocks
42(1)
2.7 Local Cyclic Cutvertices and Bridges
43(6)
2.8 Critical Blocks and Cycle Connectivity
49(3)
2.9 t-Level Components of Fuzzy Graphs
52(3)
3 Connectivity and Wiener Indices of Fuzzy Graphs
55(38)
3.1 Connectivity Index
55(2)
3.2 Bounds for Connectivity Index
57(1)
3.3 Connectivity Index of Edge Deleted and Vertex Deleted Fuzzy Graphs
58(4)
3.4 Average Connectivity Index of a Fuzzy Graph
62(4)
3.5 Connectivity Index of Fuzzy Cycles
66(3)
3.6 Algorithms
69(2)
3.7 Wiener Index of Fuzzy Graphs
71(3)
3.8 Relationship Between WI and CI of a Fuzzy Graph
74(6)
3.9 Wiener Indices of Fuzzy Trees and Fuzzy Cycles
80(2)
3.10 Applications
82(11)
3.10.1 Human Trafficking
82(3)
3.10.2 Internet Routing
85(2)
3.10.3 Illegal Immigration
87(6)
4 Distances and Convexity in Fuzzy Graphs
93(34)
4.1 Fuzzy Geodetic Convex Sets
93(5)
4.2 Fuzzy Geodetic Blocks and Their Characterization
98(3)
4.3 Fuzzy Geodetic Boundary Vertices and Interior Vertices
101(2)
4.4 Monophonic Convexity in Fuzzy Graphs
103(2)
4.5 Fuzzy Monophonic Blocks and Their Characterization
105(2)
4.6 Fuzzy Monophonic Boundary and Interior Vertices
107(1)
4.7 g-Distance in Fuzzy Graphs
108(4)
4.8 Sum Distance in Fuzzy Graphs
112(4)
4.9 Boundary and Interior in Sum Distance
116(2)
4.10 Strong Sum Distance
118(3)
4.11 Energy of a Fuzzy Graph
121(6)
5 Aggregation Operators and f-Norm Fuzzy Graphs
127(50)
5.1 Preliminaries
128(1)
5.2 t-Norm Fuzzy Graphs
128(7)
5.3 Application
135(3)
5.4 Generalized Fuzzy Relations
138(8)
5.5 Fuzzy Equivalence Relations
146(1)
5.6 Application: Illegal Immigration to the United States Through Mexico
147(2)
5.7 Operations of t-Norm Fuzzy Graphs
149(6)
5.8 Quasi-fuzzy Graphs
155(1)
5.9 Aggregation Operators
156(2)
5.10 * Composition
158(4)
5.11 * -- ⊗ Composition
162(3)
5.12 Norm and Median Functions
165(4)
5.13 Applications
169(5)
5.13.1 Trafficking in Persons
169(1)
5.13.2 Linguistic Description
170(1)
5.13.3 Slavery
171(3)
5.14 Appendix
174(3)
5.14.1 Vulnerability
174(1)
5.14.2 Government Responses
174(3)
6 Dialectic Synthesis
177(20)
6.1 Complementary Dialectic Synthesis
178(3)
6.2 Complementary Fuzzy Dialectic Synthesis Graphs
181(9)
6.3 Application
190(7)
References 197(10)
Index 207
Dr. John N. Mordeson is Professor Emeritus of Mathematics at Creighton University. He received his B.S., M.S., and Ph.D. from Iowa State University. He is a Member of Phi Kappa Phi. He is the President of the Society for Mathematics of Uncertainty. He has published 15 books and 200 journal articles. He is on the editorial board of numerous journals. He has served as an external examiner of Ph.D. candidates from India, South Africa, Bulgaria, and Pakistan. He has refereed for numerous journals and granting agencies. He is particularly interested in applying mathematics of uncertainty to combat the problem of human trafficking.Dr. Sunil Mathew is currently a Faculty Member in the Department of Mathematics, NIT Calicut, India. He has acquired his masters from St. Josephs College Devagiri, Calicut, and Ph.D. from National Institute of Technology Calicut in the area of Fuzzy Graph Theory. He has published more than 75 research papers and written two books. He is a member of several academic bodies and associations. He is editor and reviewer of several international journals. He has an experience of 20 years in teaching and research. His current research topics include fuzzy graph theory, bio-computational modeling, graph theory, fractal geometry, and chaos.
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