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=diagd1,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=γ+1≥γ+2≥⋯≥γ+n≥0, are normalized signless laplacian eigenvalues of g. in this paper some bounds for the sum k+(g)=∑ni=1/1γ+i are considered.