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. The Ihara zeta function, originally associated to certain groups and then combinatorially

Abstract. A class of CW-complexes, called self-similar complexes, is introduced, together with C ∗-algebras Aj of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian ∆j belongs to Aj, L 2-Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincaré characteristic is proved. L 2-Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals. 1. Introduction. In this paper we address the question of the possibility of extending the definition of some L 2-invariants, like the L 2-Betti numbers and Novikov-Shubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe [29], who defined a trace

Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression. 1

We show that a near-diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on-diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off-diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near-diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel.