On the generalized characteristic polynomial of a graph

Let G be a graph on n vertices, and let A(G') and D(G') denote respectively the adjacency matrix and the degree matrix of G. The generalized characteristic polynomial of G is defined as F-G(lambda, mu) = det(lambda I-n, - (A(G) - mu D(G))), where I-n is the identity matrix of size n.. We can write F-G(lambda, mu) in the coefficient form Sigma(n)(i=0) c(mu i)(G)lambda(n-i). In this paper, we give combinatorial expressions for the first five coefficients of F-G(lambda, mu). The eigenvalues of the matrix A(G) - mu D(G) of some graphs are obtained. Furthermore, we compute the generalized characteristic polynomials for all graphs on at most 10 vertices, and count the number of such graphs for which there is another graph with the same generalized characteristic polynomial. The present data show that the generalized characteristic polynomial is quite efficient to characterize graphs.