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 Tr ( v 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 is defined as D Q ( 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. [ABSTRACT FROM AUTHOR]