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 ρ 1 ≥ ρ 2 ≥ ... ≥ ρ n ≥ 0. For 1 ≤ k ≤ n , let M k (G) = ∑ i = 1 k ρ i and N k (G) = ∑ i = 0 k − 1 ρ 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 σ (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). We also show that it holds for k = n−1 and n for all graphs and for some k for r-transmission regular graphs. [ABSTRACT FROM AUTHOR]