On Distance Signless Laplacian Spectral Radius and Distance Signless Laplacian Energy

Luis Medina,Macarena Trigo,Hans Nina

doi:10.3390/math8050792

Abstract

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs for which those bounds are attained are characterized.

Highlights

Introduction and Preliminaries on Distance MatrixLet G = (V ( G ), E( G )) be a connected simple undirected graph with vertices set V ( G ) and edges set E( G )
In [2], Aouchiche and Hansen introduced the distance signless Laplacian matrix of a connected graph G as the n × n matrix defined by DQ( G ) = Tr ( G ) + D ( G ), where D ( G ) is the distance matrix of G and Tr ( G ) is the diagonal matrix of vertex transmissions of G
In [4], Hong and You gave a lower bound on the distance signless Laplacian spectral radius in terms of the sum row of matrix

Summary

Introduction and Preliminaries on Distance Matrix

In [2], Aouchiche and Hansen introduced the distance signless Laplacian matrix of a connected graph G as the n × n matrix defined by DQ( G ) = Tr ( G ) + D ( G ), where D ( G ) is the distance matrix of G and Tr ( G ) is the diagonal matrix of vertex transmissions of G. In [4], Hong and You gave a lower bound on the distance signless Laplacian spectral radius in terms of the sum row of matrix. In [7], Alhevaz et al determined some upper and lower bounds on the distance signless Laplacian spectral radius of G based on its order and independence number and characterized the extremal graphs. Let G be a simple connected graph of order n with distance degree sequence { Tr1 , Tr2 , .

Lower Bounds on the Distance Signless Laplacian Spectral Radius

Upper Bounds on the Distance Signless Laplacian Energy

Conclusions