On Vertex Matching Polynomial of Graphs

Abstract

Let $G$ be a graph with $n$ vertices. The vertex matching polynomial $M_v(G,x)$ of the graph $G$ is defined as the sum of $(-1{)}^r{q}_v(G,r)x^{n-r}$, in which $q_v(G,r)$ is the number of $r$-vertex independent sets. In this paper, we extend some important properties of the matching polynomial to the vertex matching polynomial $M_v(G,2x)$. The matching and vertex matching polynomials of some important class of graphs and some applications in nanostructures are presented.