Graphs determined by their generalized characteristic polynomials

For a given graph G with (0, 1)-adjacency matrix A(G), the generalized characteristic polynomial of G is defined to be phi(G) = phi(G)(lambda, t) = det(lambda I - (A(G) - tD(G))), where I is the identity matrix and D-G is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial phi(G). We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by phi(G). We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching. (C) 2010 Elsevier Inc. All rights reserved.