Sharp Bounds on (Generalized) Distance Energy of Graphs

Abdollah Alhevaz, Maryam Baghipur, Kinkar Ch. Das and Yilun Shang
Additional contact information
Abdollah Alhevaz: Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
Maryam Baghipur: Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
Kinkar Ch. Das: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
Yilun Shang: Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Mathematics, 2020, vol. 8, issue 3, 1-20

Abstract: Given a simple connected graph G , let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G . We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( 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.

Keywords: distance energy; distance (signless) Laplacian energy; generalized distance energy; transmission regular graph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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