Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G) = {v₁, v₂,…, v_n}. The transmission Tr(v_i) (or D_i) of vertex v_i is defined to be the sum of distances from v_i to all other vertices. Let Tr(G) be the n × n diagonal matrix with its (i, i)-entry equal to Tr_G(v_i). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as Q(G) = Tr(G) + D(G) , where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.