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

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 Pm and the Cartesian product of graphs G and H byG×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 Cn denote the cycle with n vertices. Then Pm(C4 × T ) = ∏ (2 + 2), where the product ranges over all eigenvalues of T. Moreover, we prove that Pm(C4× T ) is always a square or double a square. 2. Let T be a tree. Then Pm(P4 × T ) = ∏ (1 + 3 2 + 4), where the product ranges over all non-negative eigenvalues of T. 3. Let T be a tree with a perfect matching. Then Pm(P3 × T ) = ∏ (2 + 2), where the product ranges over all positive eigenvalues of T. Moreover, we prove that Pm(C4 × T )= [Pm(P3 × T )]2. © 2005 Elsevier B.V. All rights reserved.