The coefficients of the reduced Bartholdi zeta function

TL;DR

This paper introduces the reduced Bartholdi zeta function for graphs, linking its coefficients to graph structures and providing formulas for their calculation.

Contribution

It establishes a new zeta function for graphs, relates its coefficients to star subgraphs, and derives a general formula for these coefficients.

Findings

Coefficients count star subgraphs in the symmetric digraph D(G)

Properties of semi-principle minors are analyzed

A general formula for coefficients is provided

Abstract

In this paper, we establish a new zeta function based on the Bartholdi zeta function for an undirected graph G called the reduced Bartholdi zeta function. We study the relation between its coefficients and the structure of the graph, and demonstrate that the coefficients count the star subgraphs in the symmetric digraph D(G). Moreover, we investigate the properties of semi principle minors extracted from the adjacency matrix of the oriented line graph of G. We also present a general formula for calculating all the coefficients of the reduced Bartholdi zeta function.