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An inequality between variable wiener index and variable szeged index

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  • Hriňáková, Katarína
  • Knor, Martin
  • Škrekovski, Riste

Abstract

A well-known inequality between the Szeged and Wiener indices says that Sz(G)=∑e=ij∈E(G)ne(i)ne(j)≥∑{u,v}d(u,v)=W(G) for every graph G. In the past, variable variations of the standard topological indices were defined. Following this line, we study a natural generalisation of the above inequality, namely ∑e=ij∈E(G)(ne(i)ne(j))α≥∑{u,v}d(u,v)α. We show that for all trees the inequality is true if α > 1, and the opposite inequality holds if 0 ≤ α < 1. In fact, the first result also holds for bipartite graphs and for graphs on n vertices with at most n+3 edges, but the opposite one does not. For general graphs we solve also the case α < 0 and we present interesting conjectures. Observe, that both the sums are interesting on their own, and in accordance with the usual terminology they can be called the variable Szeged and variable Wiener indices.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:apmaco:v:362:y:2019:i:c:44
    DOI: 10.1016/j.amc.2019.124557
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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. Andova, Vesna & Orlić, Damir & Škrekovski, Riste, 2017. "Leapfrog fullerenes and Wiener index," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 281-288.
    3. Knor, Martin & Škrekovski, Riste & Tepeh, Aleksandra, 2016. "Digraphs with large maximum Wiener index," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 260-267.
    4. Bonamy, Marthe & Knor, Martin & Lužar, Borut & Pinlou, Alexandre & Škrekovski, Riste, 2017. "On the difference between the Szeged and the Wiener index," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 202-213.
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    Cited by:

    1. Cambie, Stijn & Haslegrave, John, 2022. "On the relationship between variable Wiener index and variable Szeged index," Applied Mathematics and Computation, Elsevier, vol. 431(C).

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