Some remarks on inverse Wiener index problem

The sum of distances between all pairs of vertices W (G) in a connected graph G as a graph invariant was first introduced by Wiener [9] in 1947. He observed a correlation between boiling points of paraffins and this invariant, which has later become known as Wiener index of a graph. Today, the Wiener index is one of the most widely used descriptors in chemical graph theory. Due to its strong connection to chemistry, where molecules have a tree-like structures, a lot of research was done on acyclic graphs (see [2] for survey). In 1995 Gutman and Ye [3] considered an inverse Wiener index problem. They asked for which integers n there exist trees with Wiener index n, and posed the following conjecture: