Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians

Abstract

The Pfaffian method enumerating perfect matchings of plane graphs was discovered by Kasteleyn. We use this method to enumerate perfect matchings in a type of graphs with reflective symmetry which is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (1997) 67, MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) If G is a reflective symmetric plane graph without vertices on the symmetry axis, then the number of perfect matchings of G can be expressed by a determinant of order | G|/2, where | G| denotes the number of vertices of G. (2) If G contains no subgraph which is, after the contraction of at most one cycle of odd length, an even subdivision of K 2,3, then the number of perfect matchings of G× K 2 can be expressed by a determinant of order | G|. (3) Let G be a bipartite graph without cycles of length 4 s, s∈{1,2,…}. Then the number of perfect matchings of G× K 2 equals ∏(1+ θ 2) m θ , where the product ranges over all non-negative eigenvalues θ of G and m θ is the multiplicity of eigenvalue θ. Particularly, if T is a tree then the number of perfect matchings of T× K 2 equals ∏(1+ θ 2) m θ , where the product ranges over all non-negative eigenvalues θ of T and m θ is the multiplicity of eigenvalue θ.