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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. Spiro, Sam, 2022. "The Wiener index of signed graphs," Applied Mathematics and Computation, Elsevier, vol. 416(C).
    2. Al-Yakoob, Salem & Stevanović, Dragan, 2020. "On transmission irregular starlike trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    3. Wang, Guangfu & Liu, Yajing, 2020. "The edge-Wiener index of zigzag nanotubes," Applied Mathematics and Computation, Elsevier, vol. 377(C).

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