Séminaire Lotharingien de Combinatoire, B54m (2007), 40 pp.

Gilbert Labelle, Pierre Leroux and Martin G. Ducharme

Graph Weights Arising From Mayer's Theory of Cluster Integrals

Abstract. We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory of cluster integrals in the context of a non-ideal gas. Various choices of the interaction potential between two particles yield various graph weights w(g). For example, in the case of the Gaussian interaction, the so-called Second Mayer weight w(c) of a connected graph c is closely related to the graph complexity, i.e., the number of spanning trees, of c. We give special attention to the Second Mayer weight w(c) which arises from the hard-core continuum gas in one dimension. This weight is a signed volume of a convex polytope P(c) naturally associated with c. Among our results are the values w(c) for all 2-connected graphs c of size at most 6, in Appendix B, and explicit formulas for three infinite families: complete graphs, (unoriented) cycles and complete graphs minus an edge.

Received: January 27, 2006. Accepted: January 15, 2007. Final Version: July 3, 2007.

The following versions are available: