  We consider the Schr\"odinger operator on zigzag graphs with a periodic
potential. The spectrum of this operator consists of an absolutely continuous
part (intervals separated by gaps) plus an infinite number of eigenvalues with
infinite multiplicity. We describe all compactly supported eigenfunctions with
the same eigenvalue. We define a Lyapunov function, which is analytic on some
Riemann surface. On each sheet, the Lyapunov function has the same properties
as in the scalar case, but it has branch points, which we call resonances. We
prove that all resonances are real. We determine the asymptotics of the
periodic and anti-periodic spectrum and of the resonances at high energy. We
show that there exist two types of gaps: i) stable gaps, where the endpoints
are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps,
where the endpoints are resonances (i.e., real branch points of the Lyapunov
function). We obtain the following results from the inverse spectral theory: 1)
we describe all finite gap potentials, 2) the mapping: potential -- all
eigenvalues is a real analytic isomorphism for some class of potentials.
  We apply all these results to quasi-1D models of zigzag single-well carbon
nanotubes.
