Some weighted Bartholdi zeta function of a digraph

Related to the weighted bond scattering matrix of a graph, we define some weighted Bartholdi zeta function of a digraph D, and give a determinant expression for it. We present a decomposition formula for the weighted zeta function of a group covering of D. Furthermore, we define an L-function of D, and give a determinant expression for it. As a corollary, we express the weighted Bartholdi zeta function of a group covering of D by means of its L-functions.