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 ∨ ̇ G 2 and G 1 ⊻ G 2 , respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 . As an application, we give a sufficient and necessary condition for G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 to be Seidel integral. [ABSTRACT FROM AUTHOR]