Perfect matchings in random polyomino chain graphs

Let G be a (molecule) graph. A perfect matching, or Kekulé structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekulé structures of a graph is interest in chemistry, physics and mathematics. In this paper, we focus on the number of perfect matchings in polyomino chain graphs. Simple exact formulas are given for the expected value of the number of perfect matchings in random polyomino chain graphs and for the asymptotic behavior of this expectation. Moreover, the average value of the number of perfect matchings with respect to the set of all polyomino chain graphs with s square-cells.