Molecular graphs and the inverse Wiener index problem

In the drug design process, one wants to construct chemical compounds with certain properties. In order to establish the mathematical basis for the connections between molecular structures and physicochemical properties of chemical compounds, some so-called structure-descriptors or ”topological indices” have been put forward. Among them, the Wiener index is one of the most important. A long standing conjecture on the Wiener index ([6], [9]) states that for any positive integer n (except numbers from a given 49 element set), one can find a tree with Wiener index n. We proved this conjecture in [13] and [14]. However, more realistic molecular graphs are trees with degree ≤ 3 and the so-called hexagon type graphs. In this paper, we prove that every sufficiently large integer n is the Wiener index of some caterpillar tree with degree ≤ 3, and every sufficiently large even integer is the Wiener index of some hexagon type graph. Preprint submitted to Elsevier Science 8 January 2008