Graph theory and the statistics and dynamics of polymer chains

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Abstract
The random‐flight model for flexible, branched polymer molecules can be considered as a directed graph of (n+1) points and n vectors with an associated (n+1) ×n matrix G. The matrix incorporating entropy spring effects in the Rouse approach of describing chain dynamics is shown to be given by A=GTG. Likewise, the matrix used in the Zimm approach is given by Z=GGT. Moments of the distribution of the radius of gyration and its orthogonal components are given by the eigenvalues of F= (n+1)A−1. The distributions of the radii can be computed from these same eigenvalues or by performing integrations utilizing properties of a matrix P=GT, where T is an (n+1) ×n matrix formulated to eliminate the coordinates of one of the n+1 masses (since there are only 3n degrees of freedom in a coordinate system with its origin at the center of mass). The contribution to the mean segment density about the center of mass ρi as a function of distance r of any mass i is given by ρi= (βi/π 1/2)exp(−βi2r2), where βi2=nβ02(Σn+1j=1Qi,j2 )−1 and Qi,j are the elements of Q=P−1, and β0 is the parameter in the end‐to‐end distribution of a linear chain of n statistical segment. The total mean density of segments about the center of mass is thus given by the expression ρ=Σn+1i=1=ρi. All of the statistical and dynamic characteristics of the assembly of chains can thus be determined from its graph representation and are independent of how the elements of the graph are numbered.