IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v416y2022ics0096300321008134.html
   My bibliography  Save this article

On the extremal Sombor index of trees with a given diameter

Author

Listed:
  • Li, Shuchao
  • Wang, Zheng
  • Zhang, Minjie

Abstract

Based on elementary geometry, Gutman proposed a novel graph invariants called the Sombor index SO(G), which is defined as SO(G)=∑uv∈E(G)dG2(u)+dG2(v), where dG(u) and dG(v) denote the degree of u and v in G, respectively. It has been proved that the Sombor index could predict some physicochemical properties. In this paper, we characterize the extremal graphs with respect to the Sombor index among all the n-order trees with a given diameter. Firstly, we order the trees with respect to the Sombor index among the n-vertex trees with diameter 3. Then, we determine the largest and the second largest Sombor indices of n-vertex trees with a given diameter d≥4 and characterize the corresponding trees. Moreover, for n−d=3, we characterize the extremal n-order trees which reach from the third to the fourth (resp. the sixth, the seventh) largest Sombor indices with d=4 (resp. d=5,d≥6). For n−d≥4, we characterize the extremal n-order trees which reach from the third to the fifth (resp. the eighth, the ninth) largest Sombor indices with d=4 (resp. d=5,d≥6). As consequences, the top four n-order trees with respect to the Sombor index are characterized.

Suggested Citation

  • Li, Shuchao & Wang, Zheng & Zhang, Minjie, 2022. "On the extremal Sombor index of trees with a given diameter," Applied Mathematics and Computation, Elsevier, vol. 416(C).
    • Handle: RePEc:eee:apmaco:v:416:y:2022:i:c:s0096300321008134
    DOI: 10.1016/j.amc.2021.126731
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300321008134
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2021.126731?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. Cruz, Roberto & Monsalve, Juan & Rada, Juan, 2020. "Extremal values of vertex-degree-based topological indices of chemical trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    2. Cruz, Roberto & Gutman, Ivan & Rada, Juan, 2021. "Sombor index of chemical graphs," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    3. Shuchao Li & Licheng Zhang & Minjie Zhang, 2019. "On the extremal cacti of given parameters with respect to the difference of zagreb indices," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 421-442, August.
    4. Kinkar Chandra Das & Yilun Shang, 2021. "Some Extremal Graphs with Respect to Sombor Index," Mathematics, MDPI, vol. 9(11), pages 1-15, May.
    5. Jiang, Yisheng & Lu, Mei, 2021. "Maximal augmented Zagreb index of trees with given diameter," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    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. Chen, Meng & Zhu, Yan, 2024. "Extremal unicyclic graphs of Sombor index," Applied Mathematics and Computation, Elsevier, vol. 463(C).

    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. Das, Kinkar Chandra, 2024. "Open problems on Sombor index of unicyclic and bicyclic graphs," Applied Mathematics and Computation, Elsevier, vol. 473(C).
    2. Das, Kinkar Chandra & Gutman, Ivan, 2022. "On Sombor index of trees," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    3. Wang, Yiqiao & Zheng, Lina, 2020. "Computation on the difference of Zagreb indices of maximal planar graphs with diameter two," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    4. Cruz, Roberto & Rada, Juan & Sigarreta, José M., 2021. "Sombor index of trees with at most three branch vertices," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    5. Cruz, Roberto & Gutman, Ivan & Rada, Juan, 2021. "Sombor index of chemical graphs," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    6. Lkhagva Buyantogtokh & Batmend Horoldagva & Kinkar Chandra Das, 2020. "On reduced second Zagreb index," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 776-791, April.
    7. Zhang, Weilin & You, Lihua & Liu, Hechao & Huang, Yufei, 2021. "The expected values and variances for Sombor indices in a general random chain," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    8. Lkhagva Buyantogtokh & Batmend Horoldagva & Kinkar Chandra Das, 2022. "On General Reduced Second Zagreb Index of Graphs," Mathematics, MDPI, vol. 10(19), pages 1-18, September.
    9. Kinkar Chandra Das & Yilun Shang, 2021. "Some Extremal Graphs with Respect to Sombor Index," Mathematics, MDPI, vol. 9(11), pages 1-15, May.
    10. Shang, Yilun, 2022. "Sombor index and degree-related properties of simplicial networks," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    11. Rada, Juan & Rodríguez, José M. & Sigarreta, José M., 2024. "Sombor index and elliptic Sombor index of benzenoid systems," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    12. Sun, Xiaoling & Du, Jianwei, 2022. "On Sombor index of trees with fixed domination number," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    13. Chen, Meng & Zhu, Yan, 2024. "Extremal unicyclic graphs of Sombor index," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    14. Du, Jianwei & Sun, Xiaoling, 2024. "On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves," Applied Mathematics and Computation, Elsevier, vol. 464(C).

    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:eee:apmaco:v:416:y:2022:i:c:s0096300321008134. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.