On the sum of the Laplacian eigenvalues of a graph and Brouwer's conjecture

Abstract

For a simple graph G with n-vertices, m edges and having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0, let Sk(G)=∑i=1kμi, be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk(G)≤m+(k+12), for all k=1,2,…,n. We obtain upper bounds for Sk(G) in terms of the clique number ω, the vertex covering number τ 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)=∑i=1n|μi−d‾|, where d‾=2mn 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 τ and the clique number ω of the graph.