A vertex weighted Bartholdi zeta function for a graph

We define a vertex weighted Bartholdi zeta function of a graph G, and give a determinant expression of it. Furthermore, we define a vertex weighted Bartholdi L-function of a graph. As a corollary, we show that the vertex weighted Bartholdi zeta function of a regular covering of G is a product of vertex weighted Bartholdi L-functions of G.