On the roots of maximal matching polynomials

A maximal matching of a graph G is a matching of G which is not properly contained in any other matching of G. Let $m_k\left(G\right)$ be the number of maximal matchings of size k in G. The maximal matching polynomial of G is defined by $m(G;x)={\sum }_k{m}_k\left(G\right)x^k$. It is known that maximal matching polynomials generalize the well-known matching polynomials, as the matching polynomial of every graph can be obtained from the maximal matching polynomial of some other graph. While the roots of matching polynomials have been extensively studied and well understood, the study of the roots of maximal matching polynomials has not been developed. In this article, we study the location of the roots of these polynomials. We show that maximal matching polynomials of paths and cycles have only real roots, and provide interlacing relations for their roots. On the other hand, unlike matching polynomials, maximal matching polynomials can have non-real roots, and we provide an infinite family of graphs whose maximal matching polynomials have non-real roots.