On the characteristic polynomial of a special class of graphs and spectra of balanced trees

Let H be a simple graph with n vertices and G be a sequence of n rooted graphs G(1), G(2), ..., G(n). Godsil and McKay [C.D. Godsil, B.D. McKay, A new graph product and its spectrum, Bull. Austral. Math. Soc. 18 (1978) 21-28] defined the rooted product H(G), of H by G by identifying the root vertex of G(i) with the ith vertex of H, and determined the characteristic polynomial of H(G). In this paper we prove a general result on the determinants of some special matrices and. as a corollary, determine the characteristic polynomials of adjacency and Laplacian matrices of H(G). Rojo and Soto [O. Rojo, R. Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl. 403 (2005) 97-117] computed the characteristic polynomials and the spectrum of adjacency and Laplacian matrices of a class of balanced trees. As an application of our results, we obtain their conclusions by a simple method. (c) 2008 Elsevier Inc. All rights reserved.