Combinatorial Theorem for Graphs on a Lattice

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Abstract
Several problems in lattice statistical mechanics, such as the spin-½ Ising problem and the monomer-dimer problem, can be formulated in terms of the p-generating function for the weak subgraphs of a regular lattice. This paper presents an algebraic transformation theorem which allows the p-generating function for the weak subgraphs of a lattice to be determined from the p-generating function for the far less numerous subset consisting of the closed weak subgraphs. This result will be especially useful in reducing the labor required to obtain exact finite series for various problems. The theorem also enables one to give a straightforward proof of the Ising susceptibility graph theorem due to Sykes.