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

TL;DR

This paper explores graphs where removing vertices often does not change the Wiener index, presenting an infinite series of such graphs with about half of the vertices having this property.

Contribution

It introduces a relaxed problem related to \

Findings

Constructed an infinite series of graphs with nearly half of the vertices maintaining the Wiener index upon removal.

Demonstrated that the proportion of vertices with this property can approach 50%.

Extended the understanding of \

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 $\frac{1}{2}$.