Matchings and entropies of cylinders

The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has applications in statistical physics. A graph $G$ is said to be $n$-rotation symmetric if the cyclic group of order $n$ is a subgroup of the automorphism group of $G$. Jockusch (Perfect matchings and perfect squares, J. Combin. Theory Ser. A, 67(1994), 100-115) and Kuperberg (An exploration of the permanent-determinant method, Electron. J. Combin., 5(1998), #46) proved independently that if $G$ is a plane bipartite graph of order $N$ with $2n$-rotation symmetry, then the number of perfect matchings of $G$ can be expressed as the product of $n$ determinants of order $N/2n$. In this paper we give this result a new presentation. We use this result to compute the entropy of a bulk plane bipartite lattice with $2n$-notation symmetry. We obtain explicit expressions for the numbers of perfect matchings and entropies for two types of cylinders. Using the results on the entropy of the torus obtained by Kenyon, Okounkov, and Sheffield (Dimers and amoebae, Ann. Math. 163(2006), 1019--1056) and by Salinas and Nagle (Theory of the phase transition in the layered hydrogen-bonded $SnCl^2\cdot 2H_2O$ crystal, Phys. Rev. B, 9(1974), 4920--4931), we show that each of the cylinders considered and its corresponding torus have the same entropy. Finally, we pose some problems.