On Brouwer's conjecture for the sum of k largest Laplacian eigenvalues of graphs

Abstract

Let G be a simple graph with n vertices. For 1≤k≤n, denote by Sk(G) the sum of k largest Laplacian eigenvalues of G. It is conjectured by Brouwer that Sk(G)≤e(G)+(k+12), where e(G) is the number of edges in G. This conjecture has been proved to be true for k=1, k=2, k=n−1, and k=n (for all graphs), and for certain classes of graphs such as trees, unicyclic graphs, and bicyclic graphs (for all k∈{1,2,…,n}). In this paper, we show that if Brouwer's conjecture is true for all graphs when k=p(1≤p≤(n−1)/2), then it is also true for all graphs when k=n−p−1, from which Brouwer's conjecture follows automatically for all graphs when k=n−2 and k=n−3. We also show that for a given graph G and its complement G‾, if Sk(G)≤e(G)+(k+12) holds for all k, then Sk(G‾)≤e(G‾)+(k+12) holds for all k as well, which implies that Brouwer's conjecture holds for the complements of trees, unicyclic graphs, and bicyclic graphs (for all k).