The virial expansion, in statistical mechanics, makes use of the sums of the Mayer weight of all 2-connected graphs on n vertices. We study the Second Mayer weight ωM(c) and the Ree-Hoover weight ωRH(c) of a 2-connected graph c which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph c. In the present work, we use the method of graph homomorphisms, to give new formulas of Mayer and Ree-Hoover weights for special infinite families of 2-connected graphs.