On the number of Kekule structures of a type of oblate rectangles (Q2914005)

A hexagonal system is a finite connected (planar) graph without cut-vertices in which every interior face is a regular hexagon. Hexagonal systems are the natural graph representations of benzenoid hydrocarbons. A perfect matching of some graph is called (in chemistry) a Kekule structure. The number of Kekule structures in various types of hexagonal systems is extensively treated by chemists. Here the attention is focused on hexagonal systems of a type of oblate rectangles (any such graph is determined by two parameters, \(m\) and \(n\), and denoted by \(R^j(m,n)\)). Let \(\#R^j(m,n)\) be the number of Kekule structures in the corresponding graph. The main result of the authors is an explicit formulae for \(\#R^j(m,2^{n+1}-2)\) (\(n\geq 0\)). By using it, they also find that its limit value is \(\log 2\), when both \(m\) and \(n\) tend to \(\infty\).