A note on Hoffman-type identities of graphs

An eigenvalue of a graph G is called main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Hoffman [A.J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963) 30-36] proved that G is a connected k-regular graph if and only if n∏i=2t(A-λiI)=∏i=2t(k-λi)·J, where I is the unit matrix and J the all-one matrix and λ1 = k, λ2, …, λt are all distinct eigenvalues of adjacency matrix A(G). In this note, some generalizations of Hoffman identity are presented by means of main eigenvalues.