PUB - Publikationen an der Universität Bielefeld

Let G be a simple graph on n vertices with adjacency matrix A and distinct eigenvalues mu(1) > mu(2) > ... > mu(m). In his paper [1], Hoffman observed that G is a connected regular graph if and only if [GRAPHICS] holds where J is the n x n all-one matrix and I is the unit matrix. We study expressions of the form Pi(i=1)(t) (A - beta(t)I) in a more general context allowing us not only to derive Hoffman's identity in a new way, but also to compute, say, the rank of Pi(i=2)(m) (A-mu(i)I) and of Pi(mui not equal - mu1) (A-mu(i)I) for arbitrary simple graphs and to derive corresponding identities for the recently introduced classes of harmonic and semiharmonic graphs. (C) 2003. Elsevier Science Ltd. All rights reserved.