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Graphs preserving Wiener index upon vertex removal

Author

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  • Knor, Martin
  • Majstorović, Snježana
  • Škrekovski, Riste

Abstract

The Wiener index W(G) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Šoltés posed the problem of finding all graphs G such that the equality W(G)=W(G−v) holds for all their vertices v. Up to now, the only known graph with this property is the cycle C11. Our main object of study is a relaxed version of this problem: Find graphs for which Wiener index does not change when a particular vertex v is removed. In an earlier paper we have shown that there are infinitely many graphs with the vertex v of degree 2 satisfying this property. In this paper we focus on removing a higher degree vertex and we show that for any k ≥ 3 there are infinitely many graphs with a vertex v of degree k satisfying W(G)=W(G−v). In addition, we solve an analogous problem if the degree of v is n−1 or n−2. Furthermore, we prove that dense graphs cannot be a solutions of Šoltes’s problem. We conclude that the relaxed version of Šoltés’s problem is rich with a solutions and we hope that this can provide an insight into the original problem of Šoltés.

Suggested Citation

  • Knor, Martin & Majstorović, Snježana & Škrekovski, Riste, 2018. "Graphs preserving Wiener index upon vertex removal," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 25-32.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:25-32
    DOI: 10.1016/j.amc.2018.05.047
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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. Wang, Guangfu & Liu, Yajing, 2020. "The edge-Wiener index of zigzag nanotubes," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    2. Spiro, Sam, 2022. "The Wiener index of signed graphs," Applied Mathematics and Computation, Elsevier, vol. 416(C).
    3. Al-Yakoob, Salem & Stevanović, Dragan, 2020. "On transmission irregular starlike trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).

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