On a conjecture for the sum of Laplacian eigenvalues

Let GG be a simple graph with nn vertices and e(G)e(G) edges. Brouwer et al. conjectured that the sum of the kk largest Laplacian eigenvalues of GG is at most e(G)+k+12, where 1≤k≤n1≤k≤n. In this paper, this conjecture is proved to be true for the following cases: connected graphs with sufficiently large kk, unicyclic graphs, bicyclic graphs and tricyclic graphs with some restrictions, forests, etc. Moreover, we show that if GG is a tree with a specified property, then the sum of the kk largest Laplacian eigenvalues of GG is at most e(G)+2k−2e(G)+2k−2, where 1≤k≤n1≤k≤n.