Enumeration of perfect matchings of a type of Cartesian products of graphs

Abstract

Let G be a graph and let Pm ( G ) denote the number of perfect matchings of G. We denote the path with m vertices by P m and the Cartesian product of graphs G and H by G × H . In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175–188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results: 1. Let T be a tree and let C n denote the cycle with n vertices. Then Pm ( C 4 × T ) = ∏ ( 2 + α 2 ) , where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm ( C 4 × T ) is always a square or double a square. 2. Let T be a tree. Then Pm ( P 4 × T ) = ∏ ( 1 + 3 α 2 + α 4 ) , where the product ranges over all non-negative eigenvalues α of T. 3. Let T be a tree with a perfect matching. Then Pm ( P 3 × T ) = ∏ ( 2 + α 2 ) , where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm ( C 4 × T ) = [ Pm ( P 3 × T ) ] 2 .