Brouwer type conjecture for the eigenvalues of distance signless Laplacian matrix of a graph. Issue 13 (3rd October 2021) - Alhevaz, A.; Baghipur, M.; Ganie, Hilal A.; Pirzada, S.
Abstract: 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.