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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]

Let G be a simple graph of order n with vertex set V (G) = { v 1 , v 2 , ... , v n } and edge set E (G). The arithmetic–geometric matrix A AG (G) = (h i j) of G is a matrix of order n defined by h i , j = 1 2 d i d j + d j d i if v i v j ∈ E (G) and 0 otherwise, where d i stands for the degree of the vertex v i in G. The arithmetic–geometric characteristic polynomial of G is the characteristic polynomial of A AG (G). The arithmetic–geometric energy ℰ AG (G) of G is the sum of absolute values of all eigenvalues of A AG (G). In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted. [ABSTRACT FROM AUTHOR]