A New [Combinatorial] Proof of the Commutativity of Matching Polynomials for Cycles

TL;DR

This paper presents a new combinatorial proof demonstrating the compositional relationship of matching polynomials for cycles and paths, confirming a conjecture about $d$-matching polynomials introduced in prior research.

Contribution

It provides a novel combinatorial proof of the compositional property of matching polynomials for cycles, validating a conjecture on $d$-matching polynomials.

Findings

Proves $ ext{C}_k ( ext{C}_n (x)) = ext{C}_{kn} (x)$ combinatorially.

Confirms $ ext{C}_{n,d} (x) = ext{P}_d ( ext{C}_n (x))$ conjecture.

Establishes functional equations relating matching polynomials of paths and cycles.

Abstract

We prove some functional equations involving the (classical) matching polynomials of path and cycle graphs and the $d$-matching polynomial of a cycle graph. A matching in a (finite) graph $G$ is a subset of edges no two of which share a vertex, and the matching polynomial of $G$ is a generating function encoding the numbers of matchings in $G$ of each size. The $d$-matching polynomial is a weighted average of matching polynomials of degree-$d$ covers, and was introduced in a paper of Hall, Puder, and Sawin. Let $\mathcal{C}_n$ and $\mathcal{P}_n$ denote the respective matching polynomials of the cycle and path graphs on $n$ vertices, and let $\mathcal{C}_{n,d}$ denote the $d$-matching polynomial of the cycle $C_n$. We give a purely combinatorial proof that $\mathcal{C}_k (\mathcal{C}_n (x)) = \mathcal{C}_{kn} (x)$ en route to proving a conjecture made by Hall: that $\mathcal{C}_{n,d} (x) = \mathcal{P}_d (\mathcal{C}_n (x))$.