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Sharp Bounds on (Generalized) Distance Energy of Graphs

Author

Listed:
  • 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)

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.

Suggested Citation

  • Abdollah Alhevaz & Maryam Baghipur & Kinkar Ch. Das & Yilun Shang, 2020. "Sharp Bounds on (Generalized) Distance Energy of Graphs," Mathematics, MDPI, vol. 8(3), pages 1-20, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:426-:d:332955
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    Citations

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    Cited by:

    1. Bilal A. Rather & Shariefuddin Pirzada & Tariq A. Naikoo & Yilun Shang, 2021. "On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n," Mathematics, MDPI, vol. 9(5), pages 1-17, February.
    2. Rather, Bilal A. & Ganie, Hilal A. & Shang, Yilun, 2023. "Distance Laplacian spectral ordering of sun type graphs," Applied Mathematics and Computation, Elsevier, vol. 445(C).

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