Abstract
For a simple graph G with n-vertices, m edges and having Laplacian eigenvalues mu(1), mu(2), ..., mu(n-1), mu(n) = 0, let S-k (G) = Sigma(k)(i=1) mu(i), be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that S-k(G) <= m + ((k+1)(2)), for all k = 1, 2, ..., n. We obtain upper bounds for S-k(G) in terms of the clique number omega, the vertex covering number tau and the diameter d of a graph G. We show that Brouwer's conjecture holds for certain classes of graphs. The Laplacian energy LE(G) of a graph G is defined as LE(G) = Sigma(n)(i=1) vertical bar mu(i) - (d) over bar vertical bar where (d) over bar = 2m/n, is the average degree of G. We obtain an upper bound for the Laplacian energy LE(G) of a graph G in terms of the number of vertices n, the number of edges m, the vertex covering number tau and the clique number omega of the graph. (C) 2016 Elsevier Inc. All rights reserved.