Bartholdi zeta functions of graph bundles having regular fibers

Abstract

As a continuation of computing the Bartholdi zeta function of a regular covering of a graph by Mizuno and Sato in J. Combin. Theory Ser. B 89 (2003) 27, we derive in this paper some computational formulae for the Bartholdi zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the fiber is a Schreier graph or it is regular and the voltages to derive the bundle or the covering lie in an Abelian group, then the formulae can be simplified. As a byproduct. the Bartholdi zeta functions of Schreier graphs, Cayley graphs and the cartesian product of a graph and a regular graph are obtained. (c) 2004 Elsevier Ltd. All rights reserved.