Biconnected Graphs and the Multivariate Virial Expansion

#### W.G. Faris

2012, v.18, Issue 3, 357-386

ABSTRACT

The same structures occur in equilibrium statistical mechanics and combinatorics. In particular, the virial expansion in statistical mechanics has a natural analog in combinatorics. In the latter setting, the graphical structures are built over colored sets, so the corresponding exponential generating functions are functions of several variables. The main results are a convergent virial expansion and a Legendre transform, both in the multivariate case. These are expressed in terms of the multivariate generating function for biconnected graphs.

Keywords: biconnected graph,irreducible graph,2-connected graph,combinatorial species,exponential generating function,dissymmetry theorem,cluster expansion,virial expansion,Legendre transform,equilibrium particle system,polymer system