Ihara zeta functions for some simple graph families

TL;DR

This paper simplifies the calculation of Ihara zeta functions for certain graph families, proves their effectiveness as invariants for rank-two graphs, and computes them for various specific graph classes.

Contribution

It introduces a simplified method for calculating Ihara zeta functions and establishes their completeness as invariants for rank-two graphs.

Findings

Zeta function is a complete invariant for rank-two graphs.

Reciprocal zeta function is an even polynomial for bipartite graphs.

Explicit zeta functions are determined for several graph families.

Abstract

The reciprocal of the Ihara zeta function of a graph is a polynomial invariant introduced by Ihara in 1966. Scott and Storm gave a method to determine the coefficients of the polynomial. Here we simplify their calculation and determine the zeta function for all graphs of rank two. We verify that it is a complete invariant for such graphs: If $G_1$ and $G_2$ are of rank two, then $G_1$ and $G_2$ are isomorphic if and only if they have the same Ihara zeta function. We observe that the reciprocal of the zeta function is an even polynomial if the graph is bipartite. We also determine the zeta function for several graph families: complete graphs, complete bipartite graphs, M\"{o}bius ladders, cocktail party graphs, and all graphs of order five or less. We use the special value $u=1$ to count the spanning trees for these families.