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

Let G = (V,E), V = {v(1),v(2),...,v(n)}, be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d(1) >= d(2) >= ... >= d(n) > 0, d(i) = d(v(i)). Let A = (a(ij))(nxn) and D = diag(d(1),d(2),...,d(n)) 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 - gamma(+)(1) >= gamma(+)(2) >= ... >= gamma(+)(n) = 0, are normalized signless Laplacian eigenvalues of G. In this paper some bounds for the sum K+(G) = Sigma(n)(i=1)1/gamma(+)(i) are considered.