Zeta Functions for Infinite Graphs and Functional Equations

Abstract

Abstract. The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed. IntroductionIn this paper, we review the main results concerning the Ihara zeta function and the Bartholdi zeta function for infinite graphs. Moreover, we propose various possible solutions to the problem of the validity of a functional equation for those zeta functions.The zeta function associated to a finite graph by Ihara, Sunada, Hashimoto and others, combines features of Riemann's zeta function, Artin L-functions, and Selberg's zeta function, and may be viewed as an analogue of the Dedekind zeta function of a number field [3,14,15,16,17,25,26]. It is defined by an Euler product over proper primitive cycles of the graph.A main result for the Ihara zeta function Z X (z) associated with a graph X, is the so called determinant formula, which shows that the inverse of this function can be written, up to a polynomial, as det(I − Az + Qz 2 ), where A is the adjacency matrix and Q is the diagonal matrix corresponding to the degree minus 1. As a consequence, for a finite graph, Z X (z) is indeed the inverse of a polynomial, hence can be extended meromorphically to the whole plane.A second main result is the fact that, for (q + 1)-regular graphs, namely graphs with degree constantly equal to (q + 1), Z X , or better its so called completion ξ X , satisfies a functional equation, namely is invariant under the transformation z → 1 qz . The first of the mentioned results has been proved for infinite (periodic or fractal) graphs in [11,10], by introducing the analytic determinant for operator 2000 Mathematics Subject Classification. 05C25; 05C38; 46Lxx; 11M41.

The Ihara Zeta Function of the Infinite Grid

“…On the other hand, Guido and Isola show in [10] that if the spectrum of the adjacency operator for X has certain gaps, then extending Z π using duality will match an analytic continuation of Z π across corresponding gaps in D X .…”

“…At the time of this writing there is no general theorem providing an analytic continuation, although the recent paper [10] gives some encouraging positive results. On the other hand, all known examples do have analytic continuations (albeit as multivalued functions) with finitely many isolated singularities.…”