Matchings and walks in graphs

The matching polynomial α( G, x ) of a graph G is a form of the generating function for the number of sets of k independent edges of G . in this paper we show that if G is a graph with vertex v then there is a tree T with vertex w such that \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha (G\backslash v, x)}}{{\alpha (G, x)}} = \frac{{\alpha (T\backslash w, x)}}{{\alpha (T, x)}}. $\end{document} This result has a number of consequences. Here we use it to prove that α( G \ v , 1/ x )/ x α( G , 1/ x ) is the generating function for a certain class of walks in G. As an application of these results we then establish some new properties of α( G, x ).