Improved results on Brouwer's conjecture for sum of the Laplacian eigenvalues of a graph

Let G be a graph with n vertices and m edges, and let Sk(G) be the sum of the k largest Laplacian eigenvalues of G. It was conjectured by Brouwer that Sk(G)≤m+(k+12) holds for 1≤k≤n. In this paper, we present several families of graphs for which Brouwer's conjecture holds, which improve some previously known results. We also establish a new upper bound on Sk(G) for split graphs, which is tight for each k∈{1,2,…,n−1} and turns out to be better than that conjectured by Brouwer.