Loosely speaking, a family of k-regular graphs is called a family of expanders if the size of the vertex sets goes to infinity, while the graphs maintain good connectedness. Such graphs are important in engineering applications such as network designs, complexity theory, derandomization, coding theory and cryptography. *Elementary Number Theory, Group Theory, and Ramanujan Graphs* is devoted to constructing the Ramanujan graphs which are a family of expanders. Moreover, these graphs provide an explicit example of an infinite family of graphs with large girth and large chromatic number. The large girth and chromatic number problem was originally solved by Erdös using the probabilistic method, but this does not provide a construction of such graphs.

The book covers a considerable amount of mathematical ground in order to construct and prove the results about the Ramanujan graphs: linear algebra (eigenvalues and spectral gaps), number theory (sums of two and four squares and quadratic reciprocity), and group theory (general linear groups and representation theory of finite groups). Along the way the reader will also see operators between L^{2} spaces, Chebyshev polynomials, the ring of quaternions, metabelian groups, and Cayley graphs. The fact that all these topics are used to prove graph theory results is what makes this book so interesting. The book is broken up into four chapters covering graph theory, number theory, group theory, and the Ramanujan graphs. In fact, the first three chapters can be read independently and each one is interesting.

The preface of the book claims that this book could be used for an undergraduate course. Based on the topics above I will let you decide if it is appropriate for an undergraduate course at your institution. I think that it would be difficult to use at most undergraduate colleges even as a senior capstone type course. On the other hand, any of the first three chapters could be used for an independent study course with undergraduates. The book would make a nice elective course for graduate students since it pulls so many topics together. If you are looking for a book for a faculty seminar that isn't too discipline-specific, this would be a good choice.

Overall, the book is a well written and stimulating book. My only complaint is that the book doesn't actually give any examples of the applications. It does give references for the applications, but a couple of pages devoted to enlightening the reader about the applications would have been worthwhile.

Thomas J. Pfaff ([email protected]) teaches at Ithaca College.