Kekule count in tubular hydrocarbons

Cylindrical aromatic compounds - carbon cages (fullerenes) and derived hydrocarbons - have become highly interesting objects of chemical research. A tubulene (tubular benzenoid) T is a hydrocarbon the carbon skeleton of which is a hexagonal system embedded in a cylinder (with its dangling bonds at both ends saturated with hydrogen atoms). The corresponding graph H is part of some regular hexagonal tessellation of the cylinder; its perfect matchings represent the possible double-bond arrangements, called Kekule structures (or patterns), of T. It is a well-established rule of thumb that, roughly speaking, the chemical stability of T increases with its 'Kekule count', i.e., with the number K of perfect matchings of H. In this paper, elementary algorithms of low (essentially linear) complexity are presented that allow K to be calculated for an arbitrary tubulene T. For some particular classes of tubulenes, recurrence relations and explicit formulae for K are developed and the asymptotic behavior of K is determined. Particularly remarkable is the fact that, in the theory of fully twisted tubulenes, quantities closely related to the Fibonacci numbers play an important role in the results.