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

## TL;DR

This paper extends the theory of the Bartholdi zeta function of a graph by introducing weighted derivatives and provides a formula relating the Kirchhoff index of regular coverings to that of the original graph.

## Contribution

It introduces weighted partial derivatives for the second weighted Bartholdi zeta function and derives a formula for the Kirchhoff index of regular coverings in terms of the base graph.

## Key findings

- Weighted derivatives of the Bartholdi zeta function are derived.
- A formula relating Kirchhoff indices of regular coverings and base graphs is established.
- Extensions of Li and Hou's results to weighted cases are provided.

## 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 $G$.