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.