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Graphs with a given diameter that maximise the Wiener index

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  • Sun, Qiang
  • Ikica, Barbara
  • Škrekovski, Riste
  • Vukašinović, Vida

Abstract

The Wiener index of a graph is one of the most recognised and very well-researched topological indices, i.e. graph theoretic invariants of molecular graphs. Nonetheless, some interesting questions remain largely unsolved despite being easy to state and comprehend. In this paper, we investigate a long-standing question raised by Plesník in 1984, namely, which graphs with a given diameter d attain the maximum value with respect to the Wiener index. Our approach to the problem is twofold – first we investigate the graphs with diameter smaller than or equal to 4, and then restrict our attention to graphs with diameter equal to n−c for c ≥ 1. Specifically, we provide a complete characterisation of sought-after graphs for 1 ≤ c ≤ 4 and solve the general case for c small enough in comparison to n. Along the way, we state some conjectures and propose an extension to our work.

Suggested Citation

  • Sun, Qiang & Ikica, Barbara & Škrekovski, Riste & Vukašinović, Vida, 2019. "Graphs with a given diameter that maximise the Wiener index," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 438-448.
    • Handle: RePEc:eee:apmaco:v:356:y:2019:i:c:p:438-448
    DOI: 10.1016/j.amc.2019.03.025
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    References listed on IDEAS

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    1. Das, Kinkar Ch. & Gutman, Ivan & Nadjafi–Arani, Mohammad J., 2015. "Relations between distance–based and degree–based topological indices," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 142-147.
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    Cited by:

    1. Wang, Guangfu & Liu, Yajing, 2020. "The edge-Wiener index of zigzag nanotubes," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    2. Yu Yang & Long Li & Wenhu Wang & Hua Wang, 2020. "On BC-Subtrees in Multi-Fan and Multi-Wheel Graphs," Mathematics, MDPI, vol. 9(1), pages 1-29, December.
    3. Al-Yakoob, Salem & Stevanović, Dragan, 2020. "On transmission irregular starlike trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    4. Wenjie Ning & Kun Wang, 2021. "On the Estrada Indices of Unicyclic Graphs with Fixed Diameters," Mathematics, MDPI, vol. 9(19), pages 1-11, September.

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