Weighted Bartholdi zeta functions of graphs

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