Bartholdi Zeta Functions of Branched Coverings of Digraphs

In this paper, we consider branched (di)graph coverings or graphs with semi-free action. A (di)graph with semi-free action of a group Γ is a (di)graph such that a sub(di)graph is fixed by Γ while its complement carries a free action. A branched regular covering of a (di)graph is a (di)graph, where vertices are either regular (free orbits) or totally ramified (fixed vertices). Deng, Sato and Wu treated the characteristic polynomial of a branched covering of digraph, where a subdigraph is an irregular covering of some digraph and its complement is totally ramified. We give a decompostion formula for the Bartholdi zeta function of a branched covering of a digraph D which treated by Deng, Sato and Wu. As a corollary, we obtain a decomposition formula for the Bartholdi zeta function of a graph having a semi-free action.

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