On the Laplacian eigenvalues of a graph and Laplacian energy

Let G be a simple graph with n vertices, m edges, maximum degree Delta, average degree (d) over bar = 2m/n, clique number omega having Laplacian eigenvalues mu 1, mu 2, ...,mu n-1, mu n = 0. For k (1 <= k <= n), let S-k(G) = Sigma(k)(i=1) mu(i) and let sigma (1 <= sigma <= n - 1) be the number of Laplacian eigenvalues greater than or equal to average degree (d) over bar. In this paper, we obtain a lower bound for S omega-1(G) and an upper bound for S sigma(G) in terms of m, Delta, sigma and clique number omega of the graph. As an application, we obtain the stronger bounds for the Laplacian energy LE(G) = Sigma(n)(i=1) vertical bar mu(i) - (d) over bar vertical bar , which improve some well known earlier bounds. (C) 2015 Elsevier Inc. All rights reserved.