Duality Relations of Graph Polynomials

TL;DR

This paper explores duality relations among various graph polynomials, introduces path-cover polynomials, and provides algorithms and formulas for computing specific graph properties, extending classical results and interpretations.

Contribution

It introduces path-cover polynomials with duality relations, offers combinatorial interpretations, and develops algorithms for computing graph polynomials and Hamiltonian paths in complex graphs.

Findings

Path-cover polynomials satisfy duality relations similar to matching polynomials.

An efficient algorithm for computing graph polynomials of cographs is proposed.

Explicit formulas for counting Hamiltonian paths and cycles in complete multipartite graphs are provided.

Abstract

The duality theorem of Lass relates the matching polynomials of a simple graph $G$ with the matching polynomials of its complement $\bar G$. In particular, this relation gives rise to Godsil's result, which offers a nice interpretation of the Lebesgue-Stieltjes integral associated with the Hermite orthogonality measure. In this work, we introduce the concept of path-cover polynomials. Similar to matching polynomials, we show that path-cover polynomials also satisfy duality relations and give combinatorial interpretations of the Lebesgue-Stieltjes integral and the inner product in the space of associated Laguerre polynomials. Similar duality relations hold for clique-cover polynomials and chromatic polynomials. As applications, we find an efficient algorithm that computes graph polynomials for cographs. We also give explicit formulas to compute the number of Hamiltonian paths and cycles in complete multipartite graphs.