The Clar Numbers of Capped Nanotubes

A nanotube is a closed carbon molecule in the shape of a capped cylinder. The Clar number of a carbon molecule is the maximum number of independent benzene rings over all possible Kekulé structures. We prove that at most two Clar chains are required on nanotube cylinders, giving lower bounds on the Clar number of nanotubes. In other words, a fully conjugated π-system running along the nanotube’s cylinder will be broken by at most two fracture lines. In [8], this double bond structure of capped nanotubes was described, but without a detailed mathematical proof that at most two Clar chains are required across a nanotube cylinder. We use this result to settle a conjecture in the case of long nanotubes. Carr, Wang and Ye proved that the Clar number for fullerenes on v vertices is bounded below by (Formula Presented) and further conjectured that the sharp lower bound is ^v(Formula Presented) [4]. We prove that this sharp lower bound holds for nanotubes of sufficient length. We also give a formula for the maximum number of vertices in the cap of a nanotube with chiral indices (n, m), where the caps are defined as in [3].