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On the Generalized Distance Energy of Graphs

Author

Listed:
  • Abdollah Alhevaz

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood P.O. Box: 316-3619995161, Iran)

  • Maryam Baghipur

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood P.O. Box: 316-3619995161, Iran)

  • Hilal A. Ganie

    (Department of Mathematics, University of Kashmir, Srinagar 190006, India)

  • Yilun Shang

    (Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK)

Abstract

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n .

Suggested Citation

  • Abdollah Alhevaz & Maryam Baghipur & Hilal A. Ganie & Yilun Shang, 2019. "On the Generalized Distance Energy of Graphs," Mathematics, MDPI, vol. 8(1), pages 1-16, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2019:i:1:p:17-:d:299965
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    References listed on IDEAS

    as
    1. Aouchiche, Mustapha & Hansen, Pierre, 2018. "Cospectrality of graphs with respect to distance matrices," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 309-321.
    2. Abdollah Alhevaz & Maryam Baghipur & Yilun Shang, 2019. "Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs," Mathematics, MDPI, vol. 7(10), pages 1-24, October.
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    Cited by:

    1. Abdollah Alhevaz & Maryam Baghipur & Yilun Shang, 2019. "Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs," Mathematics, MDPI, vol. 7(10), pages 1-24, October.

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