On the eigenvalues and energy of the Seidel and Seidel Laplacian matrices of graphs

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n−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 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.