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arXiv · 2311.15144

Some results on the Wiener index related to the Šoltés problem of graphs

Andrey A. Dobrynin·Konstantin V. Vorob'ev
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Abstract

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/ |V(G)|$. It is shown that there is a graph $G$ with $R_m(G) > 1/2$ for any integer $m \ge 0$. In particular, there is a regular graph of even degree with this property for any odd $m \ge 1$. The proposed approach allows to construct new families of graphs $G$ with $R_0(G) \rightarrow 1/2$ when the order of $G$ increases.

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Andrey A. Dobrynin, Konstantin V. Vorob'ev. 2023-11-26. Some results on the Wiener index related to the Šoltés problem of graphs. https://arxiv.org/abs/2311.15144

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