Decomposition Formulas of Weighted Zeta Functions of Graphs and Digraphs

We give decomposition formulas of the multiedge and the multipath zeta function of a regular covering of a graph \( G \) with respect to equivalence classes of prime, reduced cycles of \( G \). Furthermore, we give a decomposition formula of the weighted zeta function of a \( g \)-cyclic \( \Gamma \)-cover of a symmetric digraph \( D \) with respect to equivalence classes of prime cycles of \( D \), for any finite group \( \Gamma \) and \( g \in \Gamma \).