A generalized Bartholdi zeta function formula for simple graphs with bounded degree

TL;DR

This paper introduces a generalized Bartholdi zeta function for simple graphs with bounded degree, unifying and extending previous zeta functions, and provides new formulas and proofs for regular graphs.

Contribution

It develops a unified Bartholdi zeta function for bounded degree graphs and establishes a Bartholdi type formula, also offering a new heat kernel expression for regular graphs.

Findings

Unified Bartholdi zeta function for bounded degree graphs

Derived a Bartholdi type formula for these graphs

Provided a new heat kernel expression for regular graphs

Abstract

We introduce a generalized Bartholdi zeta function for simple graphs with bounded degree. This zeta function is a generalization of both the Bartholdi zeta function which was introduced by L.~Bartholdi and the Ihara zeta function which was introduced by G.~Chinta, J.~Jorgenson and A.~Karlsson. Furthermore, we establish a Bartholdi type formula of this Bartholdi zeta function for simple graphs with bounded degree. Moreover, for regular graphs, we give a new expression of the heat kernel which is regarded as a one-parameter deformation of the expression obtained by G.~Chinta, J.~Jorgenson and A.~Karlsson. By applying this formula, we give an alternative proof of the Bartholdi zeta function formula for regular graphs.