The Partial differential coefficients for the second weghted Bartholdi zeta function of a graph

Abstract

We consider the second weighted Bartholdi zeta function of a graph $G$, and present weighted versions for the result of Li and Hou's on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of $G$. Furthermore, we give a formula for the weighted Kirchhoff index of a regular covering of $G$ in terms of that of $G$.