Sharp Bounds on (Generalized) Distance Energy of Graphs

Alhevaz, Abdollah, Baghipur, Maryam, Das, Kinkar Ch. and Shang, Yilun (2020) Sharp Bounds on (Generalized) Distance Energy of Graphs. Mathematics, 8 (3). p. 426. ISSN 2227-7390

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Official URL: https://doi.org/10.3390/math8030426

Abstract

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.

Item Type: Article
Uncontrolled Keywords: distance energy; distance (signless) Laplacian energy; generalized distance energy; transmission regular graph
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Computer and Information Sciences
Depositing User: Elena Carlaw
Date Deposited: 17 Mar 2020 16:00
Last Modified: 17 Mar 2020 16:00
URI: http://nrl.northumbria.ac.uk/id/eprint/42516

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