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

Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. Theeigenvalues of matrix $mathcal{L}^{+}(G)$, $2=gamma _{1}^{+}geq gamma_{2}^{+}geq cdots geq gamma _{n}^{+}geq 0$, are normalized signlessLaplacian eigenvalues of $G$. In this paper some bounds for the sum $%K^{+}(G)=sum_{i=1}^nfrac{1}{gamma _{i}^{+}}$ are considered.