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

Abstract

Let SΓ be a Seidel matrix of a graph Γ of order n and let DΓ=diagn−1−2d1,n−1−2d2,…,n−1−2dn be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SLΓ=DΓ−SΓ. 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.