%0 Journal Article %1 godsil81 %A Godsil, C. D. %D 1981 %I Wiley %J Journal of Graph Theory %K graph.theory matching polynomial random.walk walk %N 3 %P 285--297 %R 10.1002/jgt.3190050310 %T Matchings and Walks in Graphs %V 5 %X 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 articleemptydocument$ (Gv, x)(G, x) = (Tw, x)(T, x). $documentThis result has a number of consequences. Here we use it to prove that α(G\v, 1/x)/xα(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).

@article{godsil81, abstract = {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)/xα(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).}, added-at = {2015-04-06T16:05:11.000+0200}, author = {Godsil, C. D.}, biburl = {https://www.bibsonomy.org/bibtex/22dd9329e6af3d738ce8c7a99a23f7243/ytyoun}, doi = {10.1002/jgt.3190050310}, interhash = {1effc3db2997c441386d3d51461311c3}, intrahash = {2dd9329e6af3d738ce8c7a99a23f7243}, issn = {1097-0118}, journal = {Journal of Graph Theory}, keywords = {graph.theory matching polynomial random.walk walk}, number = 3, pages = {285--297}, publisher = {Wiley }, timestamp = {2017-03-16T12:15:28.000+0100}, title = {Matchings and Walks in Graphs}, volume = 5, year = 1981 }