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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]

Let G be a simple graph of order n , without isolated vertices. Denote by A = (a i j) n × n the adjacency matrix of G. Eigenvalues of the matrix A , λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n , form the spectrum of the graph G. An important spectrum-based invariant is the graph energy, defined as E (G) = ∑ i = 1 n | λ i |. The determinant of the matrix A can be calculated as det A = ∏ i = 1 n λ i . Recently, Altindag and Bozkurt [Lower bounds for the energy of (bipartite) graphs, MATCH Commun. Math. Comput. Chem.77 (2017) 9–14] improved some well-known bounds on the graph energy. In this paper, several inequalities involving the graph invariants E (G) and | det A | are derived. Consequently, all the bounds established in the aforementioned paper are improved. [ABSTRACT FROM AUTHOR]

Let G be an undirected simple and connected graph with n vertices .n ≥ 3/ and m edges. Denote by μ1 ≥ μ2 ≥ ... ≥ μn-1 > μn = 0, γ1 ≥ γ2 ≥ ... ≥ γn, and ρ1 ≥ ρ2 ≥ ... ≥ ρn-1 > ρn = 0, respectively, the Laplacian, signless Laplacian, and normalized Laplacian eigenvalues of G. The Laplacian energy, signless Laplacian energy, and normalized Laplacian energy of G are defined as LE = Σn i=1/μi - 2m/n/, SLE = Σni=1/γi - 2m/n/, and NLE =Σni=1/ρi -1/, respectively. Lower bounds for LE, SLE, and NLE are obtained. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n and size m, the Laplacian energy of G is defined as LE(G) = where µ1,µ2,..., µn-1; µn are the Laplacian eigenvalues of G satisfying µ1 ≥ µ2 ≥ ... ≥ µn-1 > µn = 0. In this note, some new lower bounds on the graph invariant LE(G) are derived. The obtained results are compared with some already known lower bounds of LE(G). [ABSTRACT FROM AUTHOR]