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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