The Seidel spectrum of two variants of join operations

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs G(1) and G(2), let us denote the subdivision-vertex join and subdivision-edge join by G(1)(S) <(boolean OR)over dot> G(2) and G(1)(S) (boolean OR) under bar G(2), respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G(1)(S) <(boolean OR)over dot> G(2) and G(1)(S) (boolean OR) under bar G(2). As an application, we give a sufficient and necessary condition for G(1)(S) <(boolean OR)over dot> G(2) and G(1)(S) (boolean OR) under bar G(2) to be Seidel integral.