TY - JOUR

T1 - Sharp bounds on (generalized) distance energy of graphs

AU - Alhevaz, Abdollah

AU - Baghipur, Maryam

AU - Das, Kinkar Ch.

AU - Shang, Yilun

PY - 2020/3/16

Y1 - 2020/3/16

N2 - Given a simple connected graph G, let D(G) be the distance matrix, DL(G) be the distance Laplacian matrix, DQ(G) be the distance signless Laplacian matrix, and Tr(G) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . Noting that D0(G)=D(G),2D12(G)=DQ(G),D1(G)=Tr(G) and Dα(G)−Dβ(G)=(α−β)DL(G) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.

AB - Given a simple connected graph G, let D(G) be the distance matrix, DL(G) be the distance Laplacian matrix, DQ(G) be the distance signless Laplacian matrix, and Tr(G) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . Noting that D0(G)=D(G),2D12(G)=DQ(G),D1(G)=Tr(G) and Dα(G)−Dβ(G)=(α−β)DL(G) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.

KW - Distance (signless) laplacian energy

KW - Distance energy

KW - Generalized distance energy

KW - Transmission regular graph

UR - http://www.scopus.com/inward/record.url?scp=85082403795&partnerID=8YFLogxK

U2 - 10.3390/math8030426

DO - 10.3390/math8030426

M3 - Article

VL - 8

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 3

M1 - 426

ER -