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arXiv · 1810.05889

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

Garner Cochran·Corbin Groothuis·Andrew Herring·Ranjan Rohatgi·Eric Stucky
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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 $\mathcal{C}_{n,d} (x) = \mathcal{P}_d (\mathcal{C}_n (x))$.

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Garner Cochran, Corbin Groothuis, Andrew Herring, Ranjan Rohatgi, Eric Stucky. 2018-10-13. A New [Combinatorial] Proof of the Commutativity of Matching Polynomials for Cycles. https://arxiv.org/abs/1810.05889

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