Enumeration of Perfect Matchings of Graphs with Rotational Symmetry by Pfaffians

Abstract. 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. The enumeration of perfect matchings of graphs with reflective symmetry was studied extensively in the past. In this paper we consider the natural problem: how to enumerate perfect matchings of graphs with rotational symmetry? We prove 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. Furthermore, we compute the entropy of a bulk plane bipartite lattice with 2n-notation symmetry. As examples we obtain explicit expressions for the numbers of perfect matchings and entropies for two types of tilings of (the surface of) cylinders. Based on 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 · 2H2O crystal, Phys. Rev. B, 9(1974), 4920–4931), we show that each of the cylinders in our examples and its corresponding torus have the same entropy. Finally, we pose some problems.