Hoffman-type identities Original Research Article

Let G be a simple graph on n vertices with adjacency matrix A and distinct eigenvalues μ1 > μ2 > ⋯ > μm. In his paper [1], Hoffman observed that G is a connected regular graph if and only if View the MathML source holds whereJ is the n × n all-one matrix and I is the unit matrix. We study expressions of the form Πi=1t (A − ßtI) in a more general context allowing us not only to derive Hoffmanʹs identity in a new way, but also to compute, say, the rank of Πi=2m (A − μiI) and of Πμi ≠ −μ1 (A − μiI) for arbitrary simple graphs and to derive corresponding identities for the recently introduced classes of harmonic and semiharmonic graphs.