A solution to the combinatorial puzzle of Mayer’s virial expansion

Mayer’s second theorem in the context of a classical gasmodel allows us to write the coefficients of the virial expansion of pressure in terms of weighted two-connected graphs. Labelle, Leroux and Ducharme studied the graph weights arising from the one-dimensional hardcore gas model and noticed that the sum of these weights over all two-connected graphs with n vertices is –n(n–2)! . This paper addresses the question of achieving a purely combinatorial proof of this observation and extends the proof of Bernardi for the connected graph case.