Systematic generation of linear graphs - check and extension of the list of Uhlenbeck and Ford (Q752731)

Consider the problem of calculating the partition function of a system of p particles that interact via a pair potential. As is shown in every textbook on statistical mechanics [\textit{G. E. Uhlenbeck} and \textit{G. W. Ford}, ``The theory of linear graphs with applications to the theory of the visial development of the properties of gases'', Stud. Statist. Mech. 1, 119-211 (1962; Zbl 0116.455)], this problem requires the evaluation of a number of integrals each of which is in one-to-one correspondence with a linear graph with p points. If one wants to obtain an exact result for the partition function one has to evaluate all these integrals analytically. This requires, first of all, a complete list of the corresponding graphs and their invariance groups. The exact calculation of the partition function and the derivation of the resulting equation of state for clusters of hard particles was the motivation for developing an algorithm to generate a list of free graphs.