Elementary Number Theory, Group Theory and Ramanujan Graphs ( London Mathematical Society Student Texts )

Publication series ：London Mathematical Society Student Texts

Author： Giuliana Davidoff; Peter Sarnak; Alain Valette  

Publisher： Cambridge University Press‎

Publication year： 2003

E-ISBN: 9780511075551

P-ISBN(Paperback): 9780521824262

Subject： O157.5 Graph

Keyword： 图解数学、图算数学

Language： ENG

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Elementary Number Theory, Group Theory and Ramanujan Graphs

Description

This text is a self contained treatment of expander graphs and in particular their explicit construction. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering. The reader needs only a background in elementary algebra, analysis and combinatorics; the authors supply the necessary background material from graph theory, number theory, group theory and representation theory. The text can therefore be used as a brief introduction to these subjects as well as an illustration of how such topics are synthesised in modern mathematics.

Chapter

Cover

Series-title

Title

Copyright

Contents

Preface

An Overview

Chapter 1 Graph Theory

1.1. The Adjacency Matrix and Its Spectrum

Exercises on Section 1.1

1.2. Inequalities on the Spectral Gap

Exercises on Section 1.2

1.3. Asymptotic Behavior of Eigenvalues in Families of Expanders

Exercises on Section 1.3

1.4. Proof of the Asymptotic Behavior

Exercises on Section 1.4

1.5. Independence Number and Chromatic Number

Exercises on Section 1.5

1.6. Large Girth and Large Chromatic Number

Exercises on Section 1.6

1.7. Notes on Chapter 1

Chapter 2 Number Theory

2.1. Introduction

2.2. Sums of Two Squares

Exercises on Section 2.2

2.3. Quadratic Reciprocity

Exercises on Section 2.3

2.4. Sums of Four Squares

Exercises on Section 2.4

2.5. Quaternions

Exercises on Section 2.5

2.6. The Arithmetic of Integer Quaternions

Exercises on Section 2.6

2.7. Notes on Chapter 2

Chapter 3 PSL(q)

3.1. Some Finite Groups

Exercises on Section 3.1

3.2. Simplicity

Exercises on Section 3.2

3.3. Structure of Subgroups

Exercises on Section 3.3

3.4. Representation Theory of Finite Groups

Exercises on Section 3.4

3.5. Degrees of Representations of PSL(q)

Exercises on Section 3.5

3.6. Notes on Chapter 3

Chapter 4 The Graphs X

4.1. Cayley Graphs

Exercises on Section 4.1

4.2. Construction of X

Exercises on Section 4.2

4.3. Girth and Connectedness

Exercises on Section 4.3

4.4. Spectral Estimates

Exercises on Section 4.4

4.5. Notes on Chapter 4

Appendix 4-Regular Graphs with Large Girth

Exercises on Appendix

Bibliography

Index

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