The Wiener index of signed graphs

Sam Spiro

Applied Mathematics and Computation, 2022, vol. 416, issue C

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.

Keywords: Wiener index; Signed graphs; Colorings (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:416:y:2022:i:c:s0096300321008377

DOI: 10.1016/j.amc.2021.126755

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