On the sum of signless Laplacian eigenvalues of a graph

For a simple graph G, let e(G) denote the number of edges and Sk(G) denote the sum of the k largest eigenvalues of the signless Laplacian matrix of G. We conjecture that for any graph G with n vertices, for k=1,…,n. We prove the conjecture for k=2 for any graph, and for all k for regular graphs. The conjecture is an analogous to a conjecture by A.E. Brouwer with a similar statement but for the eigenvalues of Laplacian matrices of graphs.