# Expander Families and Cayley Graphs: A Beginner's Guide

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## Overview

Expander Families and Cayley Graphs: A Beginner's Guide by Mike Krebs, Anthony Shaheen

The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more.

Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive introduction to expander graphs, designed to act as a bridge between classroom study and active research in the field of expanders. It equips those with little or no prior knowledge with the skills necessary to both comprehend current research articles and begin their own research. Central to this book are four invariants that measure the quality of a Cayley graph as a communications network-the isoperimetric constant, the second-largest eigenvalue, the diameter, and the Kazhdan constant. The book poses and answers three core questions: How do these invariants relate to one another? How do they relate to subgroups and quotients? What are their optimal values/growth rates?

Chapters cover topics such as:

· Graph spectra
· A Cheeger-Buser-type inequality for regular graphs
· Group quotients and graph coverings
· Subgroups and Schreier generators
· Ramanujan graphs and the Alon-Boppana theorem
· The zig-zag product and its relation to semidirect products of groups
· Representation theory and eigenvalues of Cayley graphs
· Kazhdan constants

The only introductory text on this topic suitable for both undergraduate and graduate students, Expander Families and Cayley Graphs requires only one course in linear algebra and one in group theory. No background in graph theory or representation theory is assumed. Examples and practice problems with varying complexity are included, along with detailed notes on research articles that have appeared in the literature. Many chapters end with suggested research topics that are ideal for student projects.

## Product Details

ISBN-13: 9780199767113 Oxford University Press, USA 10/21/2011 New Edition 288 6.30(w) x 9.40(h) x 0.80(d)

Mike Krebs and Anthony Shaheen are faculty in the mathematics department at California State University, Los Angeles (CSULA). They have developed and taught a course using a draft of this book for a text, and have conducted many student research projects on expander families.

Preface
Notations and conventions
Introduction

Part 1. Basics
Chapter 1. Graph eigenvalues and the isoperimetric constant
Chapter 2. Subgroups and quotients
Chapter 3. The Alon-Boppana theorem

Part 2. Combinatorial techniques
Chapter 4. Diameters of Cayley graphs and expander families
Chapter 5. Zig-zag products

Part 3. Representation-theoretic techniques
Chapter 6. Representations of Finite Groups
Chapter 7. Representation theory and eigenvalues of Cayley graphs
Chapter 8. Kazhdan constants

Appendix A. Linear algebra
Appendix B. Asymptotic analysis of functions

Bibliography
Index

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