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The Wiener index of signed graphs

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  • Spiro, Sam

Abstract

The Wiener index of a graph W(G) is a well studied topological index for graphs. An outstanding problem of Šoltés is to find graphs G such that W(G)=W(G−v) for all vertices v∈V(G), with the only known example being G=C11. We relax this problem by defining a notion of Wiener indices for signed graphs, which we denote by Wσ(G), and under this relaxation we construct many signed graphs such that Wσ(G)=Wσ(G−v) for all v∈V(G). This ends up being related to a problem of independent interest, which asks when it is possible to 2-color the edges of a graph G such that there is a path between any two vertices of G which uses each color the same number of times. We briefly explore this latter problem, as well as its natural extension to r-colorings.

Suggested Citation

  • Spiro, Sam, 2022. "The Wiener index of signed graphs," Applied Mathematics and Computation, Elsevier, vol. 416(C).
    • Handle: RePEc:eee:apmaco:v:416:y:2022:i:c:s0096300321008377
    DOI: 10.1016/j.amc.2021.126755
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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 & Majstorović, Snježana & Škrekovski, Riste, 2018. "Graphs preserving Wiener index upon vertex removal," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 25-32.
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    Cited by:

    1. Sun, Daoqiang & Li, Long & Liu, Kai & Wang, Hua & Yang, Yu, 2022. "Enumeration of subtrees of planar two-tree networks," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    2. Guo, Songlin & Wang, Wei & Wang, Chuanming, 2023. "Disproof of a conjecture on the minimum Wiener index of signed trees," Applied Mathematics and Computation, Elsevier, vol. 439(C).

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