Some graph properties determined by edge zeta functions

TL;DR

This paper explores how the edge zeta function of a graph can determine various graph properties such as clique number, Hamiltonian cycles, and perfection, highlighting its computational potential and limitations.

Contribution

It demonstrates the ability of the edge zeta function to determine key graph properties and provides new insights into its computational capabilities and limitations.

Findings

Edge zeta function determines clique number and Hamiltonian cycles.

It can identify if a graph is perfect or chordal.

The Ihara zeta function cannot always determine these properties.

Abstract

Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal. Actually computing these properties takes exponential time. Finally, we present a new example illustrating that the Ihara zeta function cannot necessarily do the same.