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9780521531436

Elementary Number Theory, Group Theory and Ramanujan Graphs

by
  • ISBN13:

    9780521531436

  • ISBN10:

    0521531438

  • Format: Paperback
  • Copyright: 2003-01-20
  • Publisher: Cambridge University Press

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Summary

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.

Table of Contents

Preface ix
An Overview 1(7)
Graph Theory
8(30)
The Adjacency Matrix and Its Spectrum
8(4)
Inequalities on the Spectral Gap
12(6)
Asymptotic Behavior of Eigenvalues in Families of Expanders
18(2)
Proof of the Asymptotic Behavior
20(10)
Independence Number and Chromatic Number
30(2)
Large Girth and Large Chromatic Number
32(4)
Notes on Chapter 1
36(2)
Number Theory
38(34)
Introduction
38(1)
Sums of Two Squares
39(9)
Quadratic Reciprocity
48(4)
Sums of Four Squares
52(5)
Quaternions
57(2)
The Arithmetic of Integer Quaternions
59(11)
Notes on Chapter 2
70(2)
PSL2(q)
72(36)
Some Finite Groups
72(1)
Simplicity
73(3)
Structure of Subgroups
76(9)
Representation Theory of Finite Groups
85(17)
Degrees of Representations of PSL2(q)
102(5)
Notes on Chapter 3
107(1)
The Graphs Xp,q
108(24)
Cayley Graphs
108(4)
Construction of Xp,q
112(3)
Girth and Connectedness
115(7)
Spectral Estimates
122(8)
Notes on Chapter 4
130(2)
Appendix 4. Regular Graphs with Large Girth 132(6)
Bibliography 138(5)
Index 143

Supplemental Materials

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

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