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An inequality between the edge-Wiener index and the Wiener index of a graph

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  • Knor, Martin
  • Škrekovski, Riste
  • Tepeh, Aleksandra

Abstract

The Wiener index W(G) of a connected graph G is defined to be the sum ∑u, vd(u, v) of distances between all unordered pairs of vertices in G. Similarly, the edge-Wiener index We(G) of G is defined to be the sum ∑e, fd(e, f) of distances between all unordered pairs of edges in G, or equivalently, the Wiener index of the line graph L(G). Wu (2010) showed that We(G) ≥ W(G) for graphs of minimum degree 2, where equality holds only when G is a cycle. Similarly, in Knor et al. (2014), it was shown that We(G)≥δ2−14W(G) where δ denotes the minimum degree in G. In this paper, we extend/improve these two results by showing that We(G)≥δ24W(G) with equality satisfied only if G is a path on 3 vertices or a cycle. Besides this, we also consider the upper bound for We(G) as well as the ratio We(G)W(G). We show that among graphs G on n vertices We(G)W(G) attains its minimum for the star.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:714-721
    DOI: 10.1016/j.amc.2015.07.050
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    References listed on IDEAS

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    1. Li, Xueliang & Qin, Zhongmei & Wei, Meiqin & Gutman, Ivan & Dehmer, Matthias, 2015. "Novel inequalities for generalized graph entropies – Graph energies and topological indices," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 470-479.
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    Cited by:

    1. Andova, Vesna & Orlić, Damir & Škrekovski, Riste, 2017. "Leapfrog fullerenes and Wiener index," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 281-288.
    2. 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.
    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. Črepnjak, Matevž & Tratnik, Niko, 2017. "The Szeged index and the Wiener index of partial cubes with applications to chemical graphs," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 324-333.

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