Analysis on Graphs and Fractals

Ihara zeta function for infinite graphs

T Isola, Univ. Roma "Tor Vergata", Italy

Abstract

This is joint work with Daniele Guido (Univ. Roma Tor Vergata) and Michel L. Lapidus (Univ. California, Riverside) Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. In recent years infinite graphs have also been considered. First Grigorchuk and Zuk, then Clair and Mokhtari-Sharghi, have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In my talk, after reviewing the above contributions, I will mainly focus on a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.