Wiener index and graphs, almost half of whose vertices satisfy Šoltés property

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, Šoltés formulated the problem of finding all graphs G such that for every vertex v the equality 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 1 2 .