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Vertex-based and edge-based centroids of graphs

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  • Lan, Yongxin
  • Li, Tao
  • Ma, Yuede
  • Shi, Yongtang
  • Wang, Hua

Abstract

The sum of distances between all pairs of vertices, better known as the Wiener index for its applications in Chemistry, has been extensively studied in the past decades. One of the most important properties related to distance between vertices, in the form of the middle part of a tree called the “centroid”, has been thoroughly analyzed. Also arised in the study of Chemical Graph Theory is the edge Wiener index which studies the distances between edges. Various problems on this concept have been proposed and investigated, along with its correlation to the original Wiener index. We extend the study to the middle part of a tree in this note, showing interesting and sometimes rather unexpected observations on the so-called “edge centroid”. We also shed some more light on the relations between these distance-based graph invariants by investigating the behaviors of different centroids and their differences. Such edge-centroids are also compared with the vertex-based analogues in both trees and graphs. This leads to challenging questions for future work in this direction.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:331:y:2018:i:c:p:445-456
    DOI: 10.1016/j.amc.2018.03.045
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    References listed on IDEAS

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    1. Yu, Guihai & Liu, Xin & Qu, Hui, 2017. "Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 287-292.
    2. Cao, Shujuan & Dehmer, Matthias & Kang, Zhe, 2017. "Network Entropies Based on Independent Sets and Matchings," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 265-270.
    3. Lei, Hui & Li, Tao & Ma, Yuede & Wang, Hua, 2018. "Analyzing lattice networks through substructures," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 297-314.
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    Cited by:

    1. Dobrynin, Andrey A., 2019. "Infinite family of 2-connected transmission irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 1-4.
    2. 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.
    3. 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.
    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. 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.
    6. Yu, Guihai & Qu, Hui, 2018. "The coefficients of the immanantal polynomial," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 38-44.
    7. 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.
    8. 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.
    9. Tratnik, Niko, 2018. "On the Steiner hyper-Wiener index of a graph," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 360-371.

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