Enumeration of Large Combinatorial Structures

摘要

A good approximation is found to the number of non-separable sparsely edged labelled graphs, relevant in application in statistical mechanics. The enumeration of smooth labelled graphs, where we obtain an exact form for the exponential generating function, find the differential equation it satisfies and a combinatorial interpretation of this equation and finally study the sparsely edged case. We remark on a surprisingly close relationship between the results for the sparsely edged cases of the non-separable and the smooth graphs. We report further on the enumeration of bipartite graphs, labelled and unlabelled, discussed in detail in the Second Annual Report, and finally list a few further problems under investigation.