A New [Combinatorial] Proof of the Commutativity of Matching Polynomials for 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 Cn and Pn denote the respective matching polynomials of the cycle and path graphs on n vertices, and let C n,d denote the d-matching polynomial of the cycle Cn. We give a purely combinatorial proof that C k (Cn(x)) = C kn (x) en route to proving a conjecture made by Hall: that C n,d (x) = P d (Cn(x)).