relationship between coefficients of characteristic polynomial and matching polynomial of regular graphs and its applications

Suppose g is a graph, a(g) its adjacency matrix and ψ(g,λ) = λn ... +an is the characteristic polynomial of g. the matching polynomial of g is defined as m(g,x) = m(g,0)xn – m(g,1)x^n−2 – m(g,2)x^n−4 + …, where m(g,k) is the number of k−matchings in g. in this paper, the relationship between 2k-th coefficient of the characteristic polynomial, a2k, and k-th coefficient of the matching polynomial, (−1)km(g, k), k=0,1,2,…, in a regular graph is determined. in addition, these relations for finding 5,6-matchings of fullerene graphs are applied.