A trace on fractal graphs and the Ihara zeta function

Details A trace on fractal graphs and the Ihara zeta function A trace on fractal graphs and the Ihara zeta function

2006-08-02

arxiv.org/abs/math/0608060v3

Trans. Amer. Math. Soc. 361 (2009), no. 6, 3041-3070

Mathematics

Operator Algebras

30 pages, 5 figures. v3: minor corrections, to appear on Transactions AMS

Scientific paper

10.1090/S0002-9947-08-04702-8

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. 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 this article, we consider 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.

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