Seidel Signless Laplacian Energy of Graphs

Let S(G) be the Seidel matrix of a graph G of order n and let DS (G) = diag(n − 1 − 2d1, n − 1 − 2d2, . . . , n − 1 − 2dn) be the diagonal matrix with di denoting the degree of a vertex vi in G. The Seidel Laplacian matrix of G is defined as SL(G) = DS (G) − S(G) and the Seidel signless Laplacian matrix as SL+(G) = DS (G) + S(G). The Seidel signless Laplacian energy ESL+ (G) is defined as the sum of the absolute deviations of the eigenvalues of SL+(G) from their mean. In this paper, we establish the main properties of the eigenvalues of SL+(G) and of ESL+ (G).