Characterization of extremal graphs from distance signless Laplacian eigenvalues

Let G = (V, E) be a connected graph with vertex set V (G) = {v(1), v(2),..., v(n)} and edge set E(G). The transmission T-r(vi) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n x n diagonal matrix with its (i, i)-entry equal to Tr-G(v(i)). The distance signless Laplacian is defined as D-Q(G) = Tr(G)+ D(G), where D(G) is the distance matrix of G. Let partial derivative(1)(G) > partial derivative(2)(G) >= ... >= partial derivative(n)(G) denote the eigenvalues of distance signless Laplacian matrix of G. In this paper, we first characterize all graphs with partial derivative(n) (G)= n-2. Secondly, we characterize all graphs with partial derivative(2)(G) is an element of [n - 2, n] when n >= 11. Furthermore, we give the lower bound on partial derivative(2)(G) with independence number alpha and the extremal graph is also characterized. (C) 2016 Elsevier Inc. All rights reserved.