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). The transmission Tr(_vi) of vertex _vi is defined to be the sum of distances from _vi to all other vertices. Let Tr(G) be the n×n diagonal matrix with its (i,i)-entry equal to _TrG(_vi). The distance signless Laplacian is defined as ^DQ(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. Let _∂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. Secondly, we characterize all graphs with _∂2(G)∈[n-2,n] when n≥11. Furthermore, we give the lower bound on _∂2(G) with independence number α and the extremal graph is also characterized.