Mathematics – Combinatorics
Scientific paper
2004-01-01
Electronic Journal of Combinatorics 11 (2004), #R32
Mathematics
Combinatorics
28 pages, 11 figures. This is the full paper version of an invited plenary talk given at the FPSAC-03 Conference in Vadstena,
Scientific paper
The basic functional equations for connected and 2-connnected graphs can be traced back to the statistical physicists Mayer and Husimi. They play an essential role in establishing rigorously the virial expansion for imperfect gases. We survey this approach and inspired by these equations, we investigate the problem of enumerating some classes of connected graphs all of whose blocks are contained in a given class B. Included are the species of Husimi graphs (B = "complete graphs"), cacti (B = "unoriented cycles"), and oriented cacti (B = "oriented cycles"). For each of these, we consider the question of their labelled or unlabelled enumeration and of their molecular expansion, according (or not) to their block-size distributions.
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