Graphs whose Wiener index does not change when a specific vertex is removed

Abstract The Wiener index W ( G ) of a connected graph G is defined to be the sum of distances between all pairs of vertices in G . In 1991, Soltes studied changes of the Wiener index caused by removing a single vertex. He posed the problem of finding all graphs G so that equality W ( G ) = W ( G − v ) holds for all their vertices v . The cycle with 11 vertices is still the only known graph with this property. In this paper we study a relaxed version of this problem and find graphs which Wiener index does not change when a particular vertex v is removed. We show that there is a unicyclic graph G on n vertices with W ( G ) = W ( G − v ) if and only if n ≥ 9 . Also, there is a unicyclic graph G with a cycle of length c for which W ( G ) = W ( G − v ) if and only if c ≥ 5 . Moreover, we show that every graph G is an induced subgraph of H such that W ( H ) = W ( H − v ) . As our relaxed version is rich with solutions, it gives hope that Soltes’s problem may have also some solutions distinct from C 11 .