The number of Kekul, structures of polyominos on the torus

Let G be a (molecule) graph. A perfect matching, or Kekul, structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekul, structures of a (molecule) graph is interest in chemistry, physics and mathematics. In this paper, we focus on some polyominos on the torus and obtain the explicit expressions on the number of the Kekul, structures of them.