Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs

Let G = (V, E), V = {v1, v2, . . ., vn}, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d1 ≥ d2 ≥ · · · ≥ dn > 0, di = d(vi). Let A = (aij)n×n and D = diag(d1, d2, . . ., dn) be the adjacency and the diagonal degree matrix of G, respectively. Denote by L+(G) = D-1/2(D + A)D-1/2 the normalized signless Laplacian matrix of graph G. The eigenvalues of matrix L+(G), 2 = γ1+ ≥ γ2+ ≥ · · · ≥ γn+ ≥ 0, are normalized signless Laplacian eigenvalues of G. In this paper some bounds for the sum K+(G) = Pni=1 γi + are considered.