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Sharp Bounds on (Generalized) Distance Energy of Graphs

1
Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
2
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
3
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK
*
Authors to whom correspondence should be addressed.
Mathematics 2020, 8(3), 426; https://doi.org/10.3390/math8030426
Received: 10 February 2020 / Revised: 4 March 2020 / Accepted: 11 March 2020 / Published: 16 March 2020
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. View Full-Text
Keywords: distance energy; distance (signless) Laplacian energy; generalized distance energy; transmission regular graph distance energy; distance (signless) Laplacian energy; generalized distance energy; transmission regular graph
MDPI and ACS Style

Alhevaz, A.; Baghipur, M.; Das, K.C.; Shang, Y. Sharp Bounds on (Generalized) Distance Energy of Graphs. Mathematics 2020, 8, 426.

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