Transformation and topological reduction of cluster expansions using m-bonds

We introduce the notion of an "m-bond" and show how it may be used to manipulate the cluster expansions that describe the equilibrium properties of classical fluids. An m-bond has a constant value of -1, and its presence affects the sign and symmetry number of a graph. We further define an "m-product," which is formed by summing all graphs obtained by adding m-bonds to join field points in the (usual) product graph. It is shown that the logarithm of a sum of graphs can be written in terms of their m-products. The formalism is used to demonstrate a few well-known results concerning cluster expansions. Also, a generalization of the m-product is introduced, and with it a theorem is presented that relates graphs composed of f-fonds to those that contain both f- and (f+1)-bonds. Such "frustrated" graphs are useful in understanding approximations such as the Percus-Yevick formula, and also in performing numerical calculations.