Higher order matching polynomials and d-orthogonality

TL;DR

This paper demonstrates that higher-order matching polynomials of various graph families are d-orthogonal, generalizing classical orthogonal polynomials like Chebyshev, Hermite, and Laguerre, with combinatorial and algebraic insights.

Contribution

It establishes the d-orthogonality of higher-order matching polynomials for several graph families, extending classical polynomial orthogonality concepts.

Findings

Higher-order Chebyshev, Hermite, and Laguerre polynomials are d-orthogonal.

Sign-reversing involutions used to prove d-orthogonality.

Derived moments and generating functions for these polynomials.

Abstract

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.