Duality Relations of Graph Polynomials

TL;DR

This work develops duality relations for a family of graph polynomials by introducing path-cover, clique-cover, matching, and chromatic polynomials and proving operator-driven dualities that relate a graph to its complement. It builds a unified algebraic framework using operators like phi_pi, phi_mu, phi_xi, and phi_chi to express join graphs and to translate between polynomial families, drawing deep connections to Laguerre and Hermite polynomials and yielding combinatorial interpretations via Hamiltonian cycles. The paper also demonstrates practical applications, including an efficient O(N^2 log N) algorithm for computing graph polynomials on cographs and explicit formulas for Hamiltonian paths and cycles in complete multipartite graphs using Laguerre/Hermite machinery and Bell polynomials. Overall, the results unify several classical graph polynomials under a duality perspective and enable efficient computation and counting in important graph families.

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.

Figures (2)

• Figure 1: Complement of the Petersen graph $L(K_5)$.
• Figure 2: Paley graph of order $9$.

Theorems & Definitions (50)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• proof : The second proof of Theorem \ref{['path-main']}