The Seidel spectrum of two variants of join operations

Abstract

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs [Formula: see text] and [Formula: see text], let us denote the subdivision-vertex join and subdivision-edge join by [Formula: see text] and [Formula: see text], respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of [Formula: see text] and [Formula: see text]. As an application, we give a sufficient and necessary condition for [Formula: see text] and [Formula: see text] to be Seidel integral.