Weighted Bartholdi zeta functions of graphs

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