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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.

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