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arXiv · 2012.08786

Wiener index and graphs, almost half of whose vertices satisfy \v{S}olt\'{e}s property

Margarita Akhmejanova·Konstantin Olmezov·Aleksei Volostnov·Ilya Vorobyev·Konstantin Vorob'ev·Yury Yarovikov
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Abstract

The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, \v{S}olt\'{e}s formulated the problem of finding all graphs $G$ such that for every vertex $v$ the equation $W(G)=W(G-v)$ holds. The cycle $C_{11}$ is the only known graph with this property. In this paper we consider the following relaxation of the original problem: find a graph with a large proportion of vertices such that removing any one of them does not change the Wiener index of a graph. As the main result, we build an infinite series of graphs with the proportion of such vertices tending to $\frac{1}{2}$.

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Margarita Akhmejanova, Konstantin Olmezov, Aleksei Volostnov, Ilya Vorobyev, Konstantin Vorob'ev, Yury Yarovikov. 2020-12-16. Wiener index and graphs, almost half of whose vertices satisfy \v{S}olt\'{e}s property. https://arxiv.org/abs/2012.08786

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