Properties Determined by the Ihara Zeta Function of a Graph

Abstract

In this paper, we show how to determine several properties of a finite graph $G$ from its Ihara zeta function $Z_{G}(u)$. If $G$ is connected and has minimal degree at least 2, we show how to calculate the number of vertices of $G$. To do so we use a result of Bass, and in the case that $G$ is nonbipartite, we give an elementary proof of Bass' result. We further show how to determine whether $G$ is regular, and if so, its regularity and spectrum. On the other hand, we extend work of Czarneski to give several infinite families of pairs of non-isomorphic non-regular graphs with the same Ihara zeta function. These examples demonstrate that several properties of graphs, including vertex and component numbers, are not determined by the Ihara zeta function. We end with Hashimoto's edge matrix T. We show that any graph $G$ with no isolated vertices can be recovered from its $T$ matrix. Since graphs with the same Ihara zeta function are exactly those with isospectral T matrices, this relates again to the question of what information about $G$ can be recovered from its Ihara zeta function.