All but 49 Numbers are Wiener Indices of Trees

Abstract

The Wiener index is one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. A long standing conjecture on the Wiener index ([4, 5]) states that for any positive integer \(n\) (except numbers from a given 49 element set), one can find a tree with Wiener index \(n\). In this paper, we prove that every integer \(n>10^8\) is the Wiener index of some short caterpillar tree with at most six non-leaf vertices. The Wiener index conjecture for trees then follows from this and the computational results in [8] and [5].