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Adventures in Graph Theory 1st ed. 2017 [Hardback]

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  • Formāts: Hardback, 327 pages, height x width: 235x155 mm, weight: 847 g, 56 Illustrations, color; 24 Illustrations, black and white; XXVI, 327 p. 80 illus., 56 illus. in color., 1 Hardback
  • Sērija : Applied and Numerical Harmonic Analysis
  • Izdošanas datums: 22-Jan-2018
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3319683810
  • ISBN-13: 9783319683812
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Adventures in Graph Theory 1st ed. 2017Palielināt
  • Formāts: Hardback, 327 pages, height x width: 235x155 mm, weight: 847 g, 56 Illustrations, color; 24 Illustrations, black and white; XXVI, 327 p. 80 illus., 56 illus. in color., 1 Hardback
  • Sērija : Applied and Numerical Harmonic Analysis
  • Izdošanas datums: 22-Jan-2018
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3319683810
  • ISBN-13: 9783319683812
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783319683812.html

This textbook acts as a pathway to higher mathematics by seeking and illuminating the connections between graph theory and diverse fields of mathematics, such as calculus on manifolds, group theory, algebraic curves, Fourier analysis, cryptography and other areas of combinatorics. An overview of graph theory definitions and polynomial invariants for graphs prepares the reader for the subsequent dive into the applications of graph theory. To pique the reader’s interest in areas of possible exploration, recent results in mathematics appear throughout the book, accompanied with examples of related graphs, how they arise, and what their valuable uses are. The consequences of graph theory covered by the authors are complicated and far-reaching, so topics are always exhibited in a user-friendly manner with copious graphs, exercises, and Sage code for the computation of equations. Samples of the book’s source code can be found at github.com/springer-math/adventures-in-graph-theory.

The text is geared towards advanced undergraduate and graduate students and is particularly useful for those trying to decide what type of problem to tackle for their dissertation. This book can also serve as a reference for anyone interested in exploring how they can apply graph theory to other parts of mathematics.

Recenzijas

The book illustrates and explores connections between graph theory and other areas of combinatorics. This book is really interdisciplinary and it is a good source for mathematicians of diverse areas to understand its relations with graph theory. The text is geared towards advanced undergraduate and graduate students and is particularly useful for those trying to decide what type of problem to tackle for their dissertation. (Vilmar Trevisan, zbMATH 1406.05001, 2019)

1 Introduction: Graphs--Basic Definitions
1(40)
1.1 Graph theory
1(20)
1.1.1 Basic definitions
1(16)
1.1.2 Simple examples
17(4)
1.2 Some polynomial Invariants for graphs
21(20)
1.2.1 The Tutte polynomial of a graph
22(5)
1.2.2 The Ihara zeta function
27(2)
1.2.3 The Duursma zeta function
29(5)
1.2.4 Graph theory and mathematical blackjack
34(7)
2 Graphs and Laplacians
41(56)
2.1 Motivation
41(1)
2.2 Basic results
41(12)
2.3 The Moore-Penrose pseudoinverse
53(5)
2.4 Circulant graphs
58(8)
2.4.1 Cycle graphs
60(1)
2.4.2 Relationship to convolution operators
61(5)
2.5 Expander graphs
66(2)
2.6 Cayley graphs
68(9)
2.6.1 Cayley graphs on abelian groups
70(1)
2.6.2 Cayley graphs for non-abelian groups
71(6)
2.7 Additive Cayley graphs
77(9)
2.7.1 Cayley graphs and p-ary functions
80(6)
2.8 Graphs of group quotients
86(11)
2.8.1 Example of the Biggs-Smith graph
94(3)
3 Graphs as Manifolds
97(48)
3.1 Motivation
97(1)
3.2 Calculus on graphs
97(2)
3.3 Harmonic morphisms on a graph
99(22)
3.3.1 Matrix formulation of graph morphisms
102(4)
3.3.2 Identities for harmonic morphisms
106(12)
3.3.3 Covering maps
118(1)
3.3.4 Graph spectra for harmonic morphisms
119(1)
3.3.5 Riemann-Hurwitz formula
120(1)
3.4 G-equivariant covering graphs
121(5)
3.5 A Riemann-Roch theorem on graphs
126(9)
3.5.1 Divisors and the Jacobian
126(1)
3.5.2 Linear systems on graphs
127(1)
3.5.3 Non-special divisors
128(4)
3.5.4 The Riemann-Roch theorem for graphs
132(3)
3.6 Induced maps on divisors
135(10)
3.6.1 The pushforward map
136(1)
3.6.2 The pullback map
137(1)
3.6.3 Dimensions of linear systems and harmonic morphisms
138(7)
4 Chip-Firing Games
145(64)
4.1 Motivation
145(1)
4.2 Introduction
145(3)
4.2.1 The Laplacian
146(2)
4.3 Configurations on graphs
148(13)
4.3.1 Legal configurations
148(2)
4.3.2 Chip-firing and set-firing moves
150(4)
4.3.3 Stable, recurrent, and critical configurations
154(2)
4.3.4 Identifying critical configurations
156(2)
4.3.5 Reduced configurations
158(3)
4.4 Energy pairing on degree 0 configurations
161(4)
4.5 Equivalence classes of configurations
165(4)
4.6 Critical group of a graph
169(8)
4.6.1 The Jacobian and the Picard group
170(1)
4.6.2 Simple examples of critical groups
170(1)
4.6.3 The Smith normal form and Invariant factors
171(4)
4.6.4 Energy pairing on the critical group
175(2)
4.7 Examples of critical groups
177(10)
4.7.1 Trees
178(1)
4.7.2 Cycle graphs
178(1)
4.7.3 Complete graphs
179(1)
4.7.4 Wheel graphs
180(1)
4.7.5 Example of Clancy, Leake, and Payne
180(1)
4.7.6 Some Cayley graphs
181(5)
4.7.7 Graphs with cyclic critical groups
186(1)
4.8 Harmonic morphisms and Jacobians
187(8)
4.8.1 Example: morphism from cube to K4
190(5)
4.9 Dhar's burning algorithm
195(11)
4.9.1 Finding reduced configurations
197(1)
4.9.2 Finding a q-legal configuration equivalent to s
198(1)
4.9.3 Ordered Dhar's algorithm
199(5)
4.9.4 Merino's theorem
204(1)
4.9.5 Computing the critical group of a graph
205(1)
4.10 Application: Biggs' cryptosystem
206(3)
5 Interesting Graphs
209(36)
5.1 Biggs-Smith graph
210(2)
5.2 Brinkmann graph
212(2)
5.3 Chvatal graph
214(2)
5.4 Coxeter graph
216(1)
5.5 Desargues graph
216(2)
5.6 Durer graph
218(2)
5.7 Dyck graph
220(3)
5.8 Errera graph
223(2)
5.9 Foster graph
225(1)
5.10 Franklin graph
225(2)
5.11 Gray graph
227(2)
5.12 Grotzsch graph
229(3)
5.13 Heawood graph
232(1)
5.14 Hoffman graph
232(3)
5.15 Hoffman-Singleton graph
235(1)
5.16 Nauru graph
235(1)
5.17 Paley graphs
236(2)
5.18 Pappus graph
238(3)
5.19 Petersen graph
241(1)
5.20 Shrikhande graph
241(4)
6 Cayley Graphs of Bent Functions and Codes
245(58)
6.1 Motivation
245(1)
6.2 Introduction
246(1)
6.3 Bent functions
246(6)
6.4 Duals and regularity of bent functions
252(3)
6.5 Partial difference sets
255(2)
6.5.1 Dillon's correspondence
257(1)
6.6 Cayley graphs
257(6)
6.6.1 Strongly regular graphs
259(2)
6.6.2 Cayley graphs of bent functions
261(2)
6.7 Association schemes
263(5)
6.7.1 Adjacency rings (Bose-Mesner algebras)
263(1)
6.7.2 Schur rings
264(4)
6.8 The matrix-walk theorem
268(1)
6.9 Weighted partial difference sets
269(4)
6.10 Weighted Cayley graphs
273(9)
6.10.1 Edge-weighted strongly regular graphs
273(1)
6.10.2 Weighted partial difference sets
273(2)
6.10.3 Level curves of p-ary functions
275(1)
6.10.4 Intersection numbers
275(3)
6.10.5 Cayley graphs of p-ary functions
278(2)
6.10.6 Group actions on bent functions
280(2)
6.11 Fourier transforms and graph spectra
282(2)
6.11.1 Connected components of Cayley graphs
283(1)
6.12 Algebraic normal form
284(2)
6.13 Examples of bent functions
286(8)
6.13.1 Bent functions GF(3)2 → GF(3)
286(6)
6.13.2 Bent functions GF(3)3 → GF(3)
292(1)
6.13.3 Bent functions GF(5)2 → GF(5)
292(2)
6.14 Examples of Cayley graphs
294(5)
6.15 Cayley graphs of linear codes
299(1)
6.16 Analogs and questions
300(2)
6.17 Further reading
302(1)
Appendix A Selected Answers 303(6)
Bibliography 309(6)
Index 315(8)
Applied and Numerical Harmonic Analysis 323
David Joyner is a professor at the United States Naval Academy Mathematics Department. His research areas include error-correcting code, representation theory, and the applications of number theory to communication theory and cryptography.Caroline Grant Melles is a professor at the United States Naval Academy Mathematics Department.