Brouwer type conjecture for the eigenvalues of distance signless Laplacian matrix of a graph

Let G be a simple connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues rho(1) >= rho(2) >= ... >= rho(n) >= 0. For 1 <= k <= n, let M-k(G) = (i=1)Sigma(k) rho(i) and N-k(G) = Sigma(k-1)(i=0) rho(n-i) be respectively the sum of k-largest distance signless Laplacian eigenvalues and the sum of k-smallest distance signless Laplacian eigenvalues of G. In this paper, we obtain the bounds for M-k(G) and N-k(G) in terms of the number of vertices n and the transmission sigma(G) of the graph G. We propose a Brouwer-type conjecture for M-k(G) and show that it holds for graphs of diameter one and graphs of diameter two for all k. As a consequence, we observe that the conjecture holds for threshold graphs and split graphs (of diameter two). Wealso show that it holds for k = n-1 and n for all graphs and for some k for r-transmission regular graphs.