On the Eigenvalues and Energy of the Seidel and Seidel Laplacian Matrices of Graphs

Abstract

Let S ( Γ ) be a Seidel matrix of a graph Γ of order n and let D ( Γ ) = diag( n − 1 − 2 d 1 , n − 1 − 2 d 2 , …, n − 1 − 2 d n ) be a diagonal matrix with d i denoting the degree of a vertex v i in Γ . The Seidel Laplacian matrix of Γ is defined as SL( Γ ) = D ( Γ ) − S ( Γ ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs.