A TOPOLOGICAL APPROACH TO STATISTICS AND DYNAMICS OF CHAIN MOLECULES

A topological index W (Wiener index), which is the sum of all the edge s between all pairs of vertices in a chemical graph, is used for chara cterizing branching in random-flight chains. The chains are composed o f statistical bonds (or edges) of a length b jointing N beads (or vert ices). The mean square radius of gyration [S-2] of random-flight chain s is shown to be given by [S-2]=(b/N)W-2. On the other hand, the set o f partial differential equations describing the motion of the chains, whether linear or with any mode of branching, can be expressed by a co nnectivity 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 a ny form of branching by replacing the molecular weight or N with the W iener index W.