Determination of the Wiener Molecular Branching Index for the General Tree

摘要

Contemporary chemistry is focusing to an ever-increasing extent on the relationships between the structure of molecules and their physicochemical properties. In particular, there has been widespread usage of topological graphs and matrices for the characterization of both individual molecular species and a variety of intermolecular interactions. Our prime focus of interest here will center on the distance matrix, and more especially on its derivation for the important class of graphs commonly referred to as chemical trees. The many applications of the distance matrix, D(G), and the Wiener branching index, W(G), in chemistry are briefly outlined. W(G) is defined as one half the sum of all the entries in D(G). A recursion formula is developed enabling W(G) to be evaluated for any molecule whose graph G exists in the form of a tree. This formula, which represents the first general recursion formula for trees of any kind, is valid irrespective of the valence of the vertices of G or of the degree of branching in G. Several closed expressions giving W(G) for special classes of tree molecules are derived from the general formula. One illustrative worked example is also presented. Finally, it is shown how the presence of an arbitrary number of heteroatoms in tree-like molecules can readily be accommodated within our general formula by appropriately weighting the vertices and edges of G.