Elementary number theory, group theory, and Ramanujan graphs / Giuliana Davidoff, Peter Sarnak, Alain Valette

The purpose of this book is to describe the family of Ramanujan graphs introduced and studied by Lubotzky, Phillips, Sarnak and Margulis, and to give an elementary and self-contained exposition which demonstrates that the graphs have (the vast majority of) the desired properties.
The book is aimed at a mathematical audience who has seen a first course in abstract algebra, and perhaps a little analysis and combinatorics, and who is game for a fast-track introduction to selected topics in combinatorics, elementary number theory, and the linear representation theory of finite groups. It would make a great text for an honors or senior seminar, showing how elegantly many different areas of mathematics come together to solve a very concrete problem of broad interest and application.
After a brief overview, the text delves into graph theory, discussing issues of girth, chromatic number and spectral gap. From graph theory, it moves to number theory, discussing representations of integers as sums of two and four squares, taking advantage of arithmetic in the Gaussian integers and integral quaternions. It also develops the formulas for the number of representations in terms of divisor functions ... (MSN)

Bibliogr. p. 138-141. Index