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 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 [I. Gutman, Y. Yeh, The sum of all distances in bipartite graphs, Math. Slovaca 45 (1995) 327–334; M. Lepović, I. Gutman, A collective property of trees and chemical trees, J. Chem. Inf. Comput. Sci. 38 (1998) 823–826] states that for any positive integer nn (except numbers from a given 49 element set), one can find a tree with Wiener index nn. We proved this conjecture in [S. Wagner, A class of trees and its Wiener index, Acta Appl. Math. 91 (2) (2006) 119–132; H. Wang, G. Yu, All but 49 numbers are Wiener indices of trees, Acta Appl. Math. 92 (1) (2006) 15–20] However, more realistic molecular graphs are trees with degree ≤3≤3 and the so-called hexagon type graphs. In this paper, we prove that every sufficiently large integer nn is the Wiener index of some caterpillar tree with degree ≤3≤3, and every sufficiently large even integer is the Wiener index of some hexagon type graph.