IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v41y2021i4d10.1007_s10878-021-00724-2.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-021-00724-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-021-00724-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xiangxiang Liu & Ligong Wang & Xihe Li, 2020. "The Wiener index of hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 351-364, February.
    2. Debarun Ghosh & Ervin Győri & Addisu Paulos & Nika Salia & Oscar Zamora, 2020. "The maximum Wiener index of maximal planar graphs," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 1121-1135, November.
    3. Sharon, Jane Olive & Rajalaxmi, T.M. & Klavžar, Sandi & Rajan, R. Sundara & Rajasingh, Indra, 2021. "Transmission in H-naphtalenic nanosheet," Applied Mathematics and Computation, Elsevier, vol. 406(C).
    4. Dobrynin, Andrey A. & Sharafdini, Reza, 2020. "Stepwise transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    5. 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.
    6. Andrey A. Dobrynin & Andrei Yu Vesnin, 2019. "On the Wiener Complexity and the Wiener Index of Fullerene Graphs," Mathematics, MDPI, vol. 7(11), pages 1-17, November.
    7. Al-Yakoob, Salem & Stevanović, Dragan, 2020. "On transmission irregular starlike trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    8. Bezhaev, Anatoly Yu. & Dobrynin, Andrey A., 2021. "On quartic transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    9. Ghorbani, Modjtaba & Vaziri, Zahra, 2024. "On the Szeged and Wiener complexities in graphs," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    10. Anatoly Yu. Bezhaev & Andrey A. Dobrynin, 2022. "On Transmission Irregular Cubic Graphs of an Arbitrary Order," Mathematics, MDPI, vol. 10(15), pages 1-15, August.
    11. Klavžar, Sandi & Azubha Jemilet, D. & Rajasingh, Indra & Manuel, Paul & Parthiban, N., 2018. "General Transmission Lemma and Wiener complexity of triangular grids," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 115-122.
    12. Martin Knor & Riste Škrekovski, 2020. "Wiener Complexity versus the Eccentric Complexity," Mathematics, MDPI, vol. 9(1), pages 1-9, December.
    13. Kexiang Xu & Haiqiong Liu & Kinkar Ch. Das & Sandi Klavžar, 2018. "Embeddings into almost self-centered graphs of given radius," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1388-1410, November.
    14. Dobrynin, Andrey A., 2019. "Infinite family of 2-connected transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 1-4.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:41:y:2021:i:4:d:10.1007_s10878-021-00724-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.