Kekulé count in capped armchair nanotubes

By using the techniques of the transfer matrix, a formula for the number of Kekulé structures in capped armchair nanotubes is established. In effective, according to the symmetric aspect of the tubule, the size of the transfer matrix could be decreased. The study shows that the Kekulé counts in the capped armchair nanotubes is no less than 2(w(h−3)−1)α (as h→+∞), where h and w are the length and circumference of the tubule, respectively, and α≈0.4661−(2.3887/w). In particular, the above lower bound can be improved to be (4/3)(n/2)−w 2(w(h−3)−1)α if the tubule, in terms of graph theory, is bipartite (e.g. the boron-nitride nanotubes), where n is the total number of the vertices on its two caps. As an application, the closed expression for a type is given out and the numerical results for three types with length up to 50 are listed.