Decomposition Formulas of Weighted Zeta Functions of Graphs and Digraphs

Abstract

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 $\mathrm{\Gamma }$-cover of a symmetric digraph $D$ with respect to equivalence classes of prime cycles of $D$, for any finite group $\mathrm{\Gamma }$ and $g\in \mathrm{\Gamma }$.