Hoffman-type identities (Q1431929)

Let \(G\) be a simple graph and let \(A\) be the adjacency matrix of \(G\). Hoffman observed that \(G\) is a connected regular graph if and only if there exists a polynomial \(P(x)\) such that \(P(A) = J\), where \(J\) is the \(n\times n\) all-one matrix. In this work the authors generalize the Hoffman identity and derive corresponding identities for the recently introduced classes of harmonic and semiharmonic graphs.