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On the relation between Wiener index and eccentricity of a graph

Author

Listed:
  • Hamid Darabi

    (Esfarayen University of Technology)

  • Yaser Alizadeh

    (Hakim Sabzevari University)

  • Sandi Klavžar

    (University of Ljubljana
    University of Maribor
    Institute of Mathematics, Physics and Mechanics)

  • Kinkar Chandra Das

    (Sungkyunkwan University)

Abstract

The relation between the Wiener index W(G) and the eccentricity $$\varepsilon (G)$$ ε ( G ) of a graph G is studied. Lower and upper bounds on W(G) in terms of $$\varepsilon (G)$$ ε ( G ) are proved and extremal graphs characterized. A Nordhaus–Gaddum type result on W(G) involving $$\varepsilon (G)$$ ε ( G ) is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference $$W(T) - \varepsilon (T)$$ W ( T ) - ε ( T ) is minimized on caterpillars. An exact formula for $$W(T) - \varepsilon (T)$$ W ( T ) - ε ( T ) in terms of the radius of a tree T is obtained. A lower bound on the eccentricity of a tree in terms of its radius is also given. Two conjectures are proposed. The first asserts that the difference $$W(G) - \varepsilon (G)$$ W ( G ) - ε ( G ) does not increase after contracting an edge of G. The second conjecture asserts that the difference between the Wiener index of a graph and its eccentricity is largest on paths.

Suggested Citation

  • Hamid Darabi & Yaser Alizadeh & Sandi Klavžar & Kinkar Chandra Das, 2021. "On the relation between Wiener index and eccentricity of a graph," Journal of Combinatorial Optimization, Springer, vol. 41(4), pages 817-829, May.
    • Handle: RePEc:spr:jcomop:v:41:y:2021:i:4:d:10.1007_s10878-021-00724-2
    DOI: 10.1007/s10878-021-00724-2
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    References listed on IDEAS

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    1. P. Dankelmann & F. J. Osaye, 0. "Average eccentricity, minimum degree and maximum degree in graphs," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-16.
    2. P. Dankelmann & F. J. Osaye, 2020. "Average eccentricity, minimum degree and maximum degree in graphs," Journal of Combinatorial Optimization, Springer, vol. 40(3), pages 697-712, October.
    3. Kinkar Ch. Das & M. J. Nadjafi-Arani, 2017. "On maximum Wiener index of trees and graphs with given radius," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 574-587, August.
    4. Alizadeh, Yaser & Klavžar, Sandi, 2018. "On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 113-118.
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    Cited by:

    1. Hongfang Liu & Jinxia Liang & Yuhu Liu & Kinkar Chandra Das, 2023. "A Combinatorial Approach to Study the Nordhaus–Guddum-Type Results for Steiner Degree Distance," Mathematics, MDPI, vol. 11(3), pages 1-19, February.
    2. Alaa Altassan & Muhammad Imran & Shehnaz Akhter, 2022. "The Eccentric-Distance Sum Polynomials of Graphs by Using Graph Products," Mathematics, MDPI, vol. 10(16), pages 1-13, August.

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