I'm interested in band gaps of Single-walled Carbon Nanotubes (SWNTs).
I know that there are three kinds of SWNTs:
Electical properties of SWNTs depend on indices $(n,m)$:
Because of the symmetry and unique electronic structure of graphene, the structure of a nanotube strongly affects its electrical properties. For a given (n,m) nanotube, if n = m, the nanotube is metallic; if n − m is a multiple of 3, then the nanotube is semiconducting with a very small band gap, otherwise the nanotube is a moderate semiconductor.
However, this rule has some exceptions.
Where I can find proof of this statement or how I can prove it using symmetry of SWNTs?