The goal of this book is to guide the reader in a stroll through the garden of zeta functions of graphs. The subject arose in the late part of the twentieth century and was modelled on the zetas found in other gardens.

Number theory involves many zetas, starting with Riemann's – a necessary ingredient in the study of the distribution of prime numbers. Other zetas of interest to number theorists include the Dedekind zeta function of an algebraic number field and its analog for function fields. Many Riemann hypotheses have been formulated and a few proved. The statistics of the complex zeros of zeta have been connected with the statistics of the eigenvalues of random Hermitian matrices (the Gaussian unitary ensemble (GUE) distribution of quantum chaos). Artin L-functions are also a kind of zeta associated with a representation of a Galois group of number or function fields. We will find graph analogs of all of these.

Differential geometry has its own zeta, the Selberg zeta function, which is used to study the distribution of the lengths of prime geodesics in compact or arithmetic Riemann surfaces. There is a third zeta function, known as the Ruelle zeta function, which is associated with dynamical systems. We will look at these zetas briefly in Part I. The graph theory zetas are related to these zetas too.

In Part I we give a brief glimpse of four sorts of zeta function, to motivate the rest of the book. In fact, much of Part I is not necessary for the rest of the book. Feel free to skip all but Chapter 2 on the Ihara zeta function.