A new method for the calculation of the Wiener number of acyclic molecules

From Wiener index to molecules

In this paper we present an algorithm for the generation of molecular graphs with a given value of the Wiener index. The high number of graphs for a given value of the Wiener index is reduced thanks to the application of a set of heuristics taking into account the structural characteristics of the molecules. The selection of parameters as the interval of values for the Wiener index, the diversity and occurrence of atoms and bonds, the size and number of cycles, and the presence of structural patterns guide the processing of the heuristics generating molecular graphs with a considerable saving in computational cost. The modularity in the design of the algorithm allows it to be used as a pattern for the development of other algorithms based on different topological invariants, which allow for its use in areas of interest, say as involving combinatorial databases and screening in chemical databases.

The Wiener distance matrix for acyclic compounds and polymers

The relationship between the Wiener indices and the topological structures of alkanes is analyzed. The expressions for the Wiener distances between elements of these structures are derived, and the distance matrix is constructed for them; this matrix is naturally called the Wiener distance matrix. The expressions for the Wiener indices of polymers with units of arbitrary structure are obtained.

Upgrading the Wiener index

The Wiener index Wis the oldest molecular-graph-based structure-descriptor. It is defined as the sum of the distances of all pairs of vertices of the molecular graph G, where the distance is the number of edges in the shortest path connecting the respective vertices, and where G is the hydrogen-depleted molecular graph. This seemingly very simple topological index could be "upgraded" (a) by using as the distance the sum of the bond lengths along the shortest path, or (b) by using the Euclidean distance between the respective pairs of atoms. Each of these "upgraded" Wiener indices could be computed either (?) for the hydrogen- depleted or (?) for the hydrogen-filled molecular graph. We provide examples showing that none of the modifications (a?), (a?), (b?), (b?) yields better results than the ordinary Wiener index, and that there is a very good linear correlation between W and its "upgraded" variants.

On the relation between W'/W index, hyper-Wiener index, and Wiener numbers

It is shown analytically that the W'/W index, the hyper-Wiener index, and the Wiener number are closely related graph-theoretical invariants for acyclic structures. A general analytical expression for the hyper-Wiener index of a tree is derived too.