On Weakly Distinguishing Graph Polynomials

TL;DR

This paper investigates the properties of certain graph polynomials, demonstrating that the clique, independence, and harmonious chromatic polynomials are weakly distinguishing, meaning they cannot uniquely identify most graphs.

Contribution

It establishes that several important graph polynomials are weakly distinguishing, expanding understanding of their limitations in graph identification.

Findings

Clique polynomial is weakly distinguishing

Independence polynomial is weakly distinguishing

Harmonious chromatic polynomial is weakly distinguishing

Abstract

A univariate graph polynomial P(G;X) is weakly distinguishing if for almost all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing. Furthermore, we show that generating functions of induced subgraphs with property C are weakly distinguishing provided that C is of bounded degeneracy or tree-width. The same holds for the harmonious chromatic polynomial.