A topological approach to statistics and dynamics of chain molecules

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Abstract
A topological index W (Wiener index), which is the sum of all the edges between all pairs of vertices in a chemical graph, is used for characterizing branching in random‐flight chains. The chains are composed of statistical bonds (or edges) of a length b jointing N beads (or vertices). The mean square radius of gyration 〈S2〉 of random‐flight chains is shown to be given by 〈S2〉=(b/N)2W. On the other hand, the set of partial differential equations describing the motion of the chains, whether linear or with any mode of branching, can be expressed by a connectivity matrix (K). We demonstrate that a relationship between the matrix (K) and the Wiener index is given by W=N Tr (K)−1. It follows that the whole of linear chain theory can be generalized to include any form of branching by replacing the molecular weight or N with the Wiener index W.