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

Let G   be a graph and let Pm(G)(G) denote the number of perfect matchings of G.We denote the path with m   vertices by PmPm and the Cartesian product of graphs G and H   by G×HG×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 CnCn denote the cycle with n   vertices. Then Pm(C4×T)=∏(2+α2)(C4×T)=∏(2+α2), where the product ranges over all eigenvalues αα of T  . Moreover, we prove that Pm(C4×T)(C4×T) is always a square or double a square.2. Let T   be a tree. Then Pm(P4×T)=∏(1+3α2+α4)(P4×T)=∏(1+3α2+α4), where the product ranges over all non-negative eigenvalues αα of T.3. Let T   be a tree with a perfect matching. Then Pm(P3×T)=∏(2+α2),(P3×T)=∏(2+α2), where the product ranges over all positive eigenvalues αα of T  . Moreover, we prove that Pm(C4×T)=[Pm(P3×T)]2(C4×T)=[Pm(P3×T)]2.