IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v361y2019icp232-245.html
   My bibliography  Save this article

Some extremal ratios of the distance and subtree problems in binary trees

Author

Listed:
  • Li, Shuchao
  • Wang, Hua
  • Wang, Shujing

Abstract

Among many topological indices of trees the sum of distances σ(T) and the number of subtrees F(T) have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the “local” versions of these invariants, σT(v) for the sum of distance from v to all other vertices and FT(v) for the number of subtrees containing v, extremal problems can be raised and studied for vertices within a tree. This leads to the concept of “middle parts” of a tree with respect to different indices. A challenging problem is to find extremal values of the ratios between graph indices and corresponding local functions at middle parts or leaves. This problem also provides new opportunities to further verify the the correlation between different indices such as σ(T) and F(T). Such extremal ratios, along with the extremal structures, were studied and compared for the distance and subtree problems for general trees (Barefoot, 1997; Székely and Wang, 2013, 2014). In this paper, this study is extended to binary trees, a class of trees with numerous practical applications in which the extremal ratio problems appear to be even more complicated. After justifying some basic properties on the distance and subtree problems in trees and binary trees, characterizations are provided for the extremal structures achieving two extremal ratios in binary trees of given order. The generalization of this work to k-ary trees is also briefly discussed. The findings are compared with the previous established extremal structures in general trees. Lastly some potential future work is mentioned.

Suggested Citation

  • Li, Shuchao & Wang, Hua & Wang, Shujing, 2019. "Some extremal ratios of the distance and subtree problems in binary trees," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 232-245.
  • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:232-245
    DOI: 10.1016/j.amc.2019.05.023
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300319304163
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2019.05.023?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. Cai, Qingqiong & Cao, Fuyuan & Li, Tao & Wang, Hua, 2018. "On distances in vertex-weighted trees," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 435-442.
    2. Lan, Yongxin & Li, Tao & Ma, Yuede & Shi, Yongtang & Wang, Hua, 2018. "Vertex-based and edge-based centroids of graphs," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 445-456.
    Full references (including those not matched with items on IDEAS)

    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. Tratnik, Niko, 2018. "On the Steiner hyper-Wiener index of a graph," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 360-371.
    2. Dobrynin, Andrey A., 2019. "Infinite family of 2-connected transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 1-4.
    3. Lan, Yongxin & Li, Tao & Wang, Hua & Xia, Chengyi, 2019. "A note on extremal trees with degree conditions," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 70-79.
    4. Li, Fengwei & Ye, Qingfang & Broersma, Hajo & Ye, Ruixuan & Zhang, Xiaoyan, 2021. "Extremality of VDB topological indices over f–benzenoids with given order," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    5. Ma, Yuede & Cao, Shujuan & Shi, Yongtang & Dehmer, Matthias & Xia, Chengyi, 2019. "Nordhaus–Gaddum type results for graph irregularities," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 268-272.
    6. Yu, Guihai & Qu, Hui, 2018. "The coefficients of the immanantal polynomial," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 38-44.
    7. Cai, Qingqiong & Cao, Fuyuan & Li, Tao & Wang, Hua, 2018. "On distances in vertex-weighted trees," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 435-442.
    8. Li, Yinkui & Gu, Ruijuan, 2018. "Bounds for scattering number and rupture degree of graphs with genus," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 329-334.

    More about this item

    Keywords

    Tree; Subtree; Wiener index; Ratio;
    All these keywords.

    Statistics

    Access and download statistics

    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:361:y:2019:i:c:p:232-245. 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.