  Prompted by recent reports on $\sqrt{3} \times \sqrt{3}$ graphene
superlattices with intrinsic inter-valley interactions, we perform
first-principles calculations to investigate the electronic properties of
periodically nitrogen-doped graphene and carbon nanotube nanostructures. In
these structures, nitrogen atoms substitute one-sixth of the carbon atoms in
the pristine hexagonal lattices with exact periodicity to form perfect
$\sqrt{3} \times \sqrt{3}$ superlattices of graphene and carbon nanotubes.
Multiple nanostructures of $\sqrt{3} \times \sqrt{3}$ graphene ribbons and
carbon nanotubes are explored, and all configurations show nonmagnetic and
metallic behaviors. The transport properties of $\sqrt{3} \times \sqrt{3}$
graphene and carbon nanotube superlattices are calculated utilizing the
non-equilibrium Green's function formalism combined with density functional
theory. The transmission spectrum through the pristine and $\sqrt{3} \times
\sqrt{3}$ armchair carbon nanotube heterostructure shows quantized behavior
under certain circumstances.
