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Leapfrog fullerenes and Wiener index

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  • Andova, Vesna
  • Orlić, Damir
  • Škrekovski, Riste

Abstract

Fullerene graphs are cubic, 3-connected planar graphs with only pentagonal and hexagonal faces. A fullerene is called a leapfrog fullerene, Le(F), if it can be constructed by a leapfrog transformation from other fullerene graph F. Here we determine the relation between the Wiener index of Le(F) and the Wiener index of the original graph F. We obtain lower and upper bounds of the Wiener index of Lei(F) in terms of the Wiener index of the original graph. As a consequence, starting with any fullerene F, and iterating the leapfrog transformation we obtain fullerenes, Lei(F), with Wiener index of order O(n2.64) and Ω(n2.36), where n is the number of vertices of Lei(F). These results disprove Hua et al. (2014) conjecture that the Wiener index of fullerene graphs on n vertices is of order Θ(n3).

Suggested Citation

  • Andova, Vesna & Orlić, Damir & Škrekovski, Riste, 2017. "Leapfrog fullerenes and Wiener index," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 281-288.
    • Handle: RePEc:eee:apmaco:v:309:y:2017:i:c:p:281-288
    DOI: 10.1016/j.amc.2017.03.043
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    References listed on IDEAS

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    1. Knor, Martin & Škrekovski, Riste & Tepeh, Aleksandra, 2016. "Some remarks on Wiener index of oriented graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 631-636.
    2. Knor, Martin & Škrekovski, Riste & Tepeh, Aleksandra, 2015. "An inequality between the edge-Wiener index and the Wiener index of a graph," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 714-721.
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    Cited by:

    1. 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.
    2. Hriňáková, Katarína & Knor, Martin & Škrekovski, Riste, 2019. "An inequality between variable wiener index and variable szeged index," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    3. Simon Brezovnik & Niko Tratnik & Petra Žigert Pleteršek, 2021. "Weighted Wiener Indices of Molecular Graphs with Application to Alkenes and Alkadienes," Mathematics, MDPI, vol. 9(2), pages 1-16, January.
    4. 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.

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