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Let G be a simple connected graph of order n with m edges. Denote by + 1 γ≥ + 2 γ≥ . . . ≥ + n γ≥ 0 the normalized signless Laplacian eigenvalues of G. In this work, we define the normalized signless Laplacian Estrada index of G as NSEE (G) = Pn i=1 e+ i: Some lower bounds on NSEE (G) are also established. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices. The Kirchhoff index of G is defined as Kf (G) = n Pn1 i=1 > n = 0 are the Laplacian eigenvalues of G. Some bounds on Kf (G) in terms of graph parameters such as the number of vertices, the number of edges, first Zagreb index, forgotten topological index, etc., are presented. These bounds improve some previously known bounds in the literature. [ABSTRACT FROM AUTHOR]

Let G = (V;E), V = fv1; v2;:::; vng, 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)nn 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 = + 1 + 2 = + n = 0, are normalized signless Laplacian eigenvalues of G. In this paper some bounds for the sum K+(G) = Pn i=1 1 + i are considered. [ABSTRACT FROM AUTHOR]