Graphs determined by their generalized characteristic polynomials

For a given graph G with (0, 1)-adjacency matrix AG, the generalized characteristic polynomial of G is defined to be ϕG=ϕG(λ,t)=det(λI-(AG-tDG)), where I is the identity matrix and DG 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 ϕG. 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 ϕG. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching.