Graphs preserving Wiener index upon vertex removal

The Wiener index W(G) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Soltes posed the problem of finding all graphs G such that the equality W(G) = W(G - v) holds for all their vertices v. Up to now, the only known graph with this property is the cycle C-11. Our main object of study is a relaxed version of this problem: Find graphs for which Wiener index does not change when a particular vertex v is removed. In an earlier paper we have shown that there are infinitely many graphs with the vertex v of degree 2 satisfying this property. In this paper we focus on removing a higher degree vertex and we show that for any k> 3 there are infinitely many graphs with a vertex v of degree k satisfying W(G) = W(G - v). In addition, we solve an analogous problem if the degree of v is n - 1 or n - 2. Furthermore, we prove that dense graphs cannot be a solutions of Soltes's problem. We conclude that the relaxed version Soltes's problem is rich with a solutions and we hope that this can provide an insight into the original problem of Soltes. (C) 2018 Elsevier Inc. All rights reserved.