num_pages: 252
description: Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book which makes connections with quantum chaos and random matrix theory plus expander/Ramanujan graphs of interest in computer science. Pitched at beginning graduate students the book will also appeal to researchers. Many well-chosen illustrations and diagrams and exercises throughout theoretical and computer-based.
language: English
title: Zeta Functions of Graphs: A Stroll through the Garden
publisher: Cambridge University Press