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 Y(G, X) = 2 + ai im-1 + ... +an is the characteristic polynomial of G. The matching polynomial of G is defined as MG,x) = m(GO)x – m(G, 1 )x-2 - m(G, 2)x -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, azko and k-th coefficient of the matching polynomial, (-1)/m(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.