Characterization of extremal graphs from distance signless Laplacian eigenvalues

Let G=(V,E) be a connected graph with vertex set V(G)={v1,v2,…,vn} and edge set E(G)E(G). The transmission Tr(vi)Tr(vi) of vertex vivi is defined to be the sum of distances from vivi to all other vertices. Let Tr(G)Tr(G) be the n×nn×n diagonal matrix with its (i,i)-entry equal to TrG(vi)TrG(vi). The distance signless Laplacian is defined as DQ(G)=Tr(G)+D(G)DQ(G)=Tr(G)+D(G), where D(G)D(G) is the distance matrix of G  . Let ∂1(G)≥∂2(G)≥⋯≥∂n(G)∂1(G)≥∂2(G)≥⋯≥∂n(G) denote the eigenvalues of distance signless Laplacian matrix of G  . In this paper, we first characterize all graphs with ∂n(G)=n−2∂n(G)=n−2. Secondly, we characterize all graphs with ∂2(G)∈[n−2,n]∂2(G)∈[n−2,n] when n≥11n≥11. Furthermore, we give the lower bound on ∂2(G)∂2(G) with independence number α and the extremal graph is also characterized.