Abstract: A celebrated result of Marcus, Spielman, and Srivas asserts that there exist bipartite Ramanujan graphs of every degree and number of vertices. In joint work with Puder and Sawin, my coauthors and I gave a significant generalization of their result. To prove our result we introduced a family of polynomials one can attach to a finite undirected graph. We call them d-matchings polynomials because when d = 1 one obtains the so-called matchings polynomial of a graph. In this talk I will define these polynomials and give some of the remarkable properties they satisfy.