LAUSR

Abstract Let G be a graph with n vertices and m edges, and let S k ( G ) be the sum of the k largest Laplacian eigenvalues of G. It was conjectured by Brouwer that S k ( G ) ≤ m + ( k + 1 2 ) 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 S k ( 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.