The graph-like state of matter—VI: Combinatorial approach to the excluded-volume perturbation

Abstract

The close connection between graphs and matrices, known since the work of Kirchhoff in the 1840's, is reflected in the literature on the perturbation expansion of the excluded-volume theory for polymer chains. The chain molecule is in a graph-like state, and the coefficients of the expansion are related to Petrie matrices. From a purely combinatorial approach to this situation, a number of new insights are derived. The Fixman partition function for an infinite chain is shown to be not only divergent but non-asymptotic. The existence of the asymptotic expansion of the chain-expansion parameter α 2 in terms of powers of the usual parameter z being assumed, it is extremely likely that these coefficients oscillate and increase rapidly with m. By means of a pre-averaging procedure (primed coefficients c′ m , the first 18 coefficients can be estimated with a desk calculator. The first coefficient c′ 1 is thus found to be 1.299 (instead of 1.333) and the higher coefficients then fall away regularly from their ‘correct’ values. As far as these are known (i.e. up to m = 3 inclusive), the plot of α 2 vs z is only little affected up to z = 0.4. The coefficient c 4 is estimated to be about −25 by two independent methods. The graph-theoretical approach relates the coefficients c m either to the number of spanning trees of so-called contracted configuration graphs, or to the sum of determinants of intersection matrices derived from Petrie matrices. The theory of Gordon and Tutte on these sums is summarised, and appropriately extended for application to the pre-averaging technique.