Elementary Number Theory, Group Theory, and Ramanujan Graphs

Cover:

Contents:

Chapter 1. Graph Theory

1.1. The Adjacency Matrix and Its Spectrum
1.2. Inequalities on the Spectral Gap
1.3. Asymptotic Behavior of Eigenvalues in Families of Expanders
1.4. Proof of the Asymptotic Behavior
1.5. Independence Number and Chromatic Number
1.6. Large Girth and Large Chromatic Number
1.7. Notes on Chapter 1

Chapter 2. Number Theory

2.1. Introduction
2.2. Sums of Two Squares
2.3. Quadratic Reciprocity
2.4. Sums of Four Squares
2.5. Quaternions
2.6. The Arithmetic of Integer Quaternions
2.7. Notes on Chapter 2

Chapter 3. PSL2(q)

3.1. Some Finite Groups
3.2. Simplicity
3.3. Structure of Subgroups
3.4. Representation Theory of Finite Groups
3.5. Degrees of Representations of PSL2(q)
3.6. Notes on Chapter 3

Chapter 4. The Graphs X p,q

4.1. Cayley Graphs
4.2. Construction of X p,q
4.3. Girth and Connectedness
4.4. Spectral Estimates
4.5. Notes on Chapter 4

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