  We prove that the two-dimensional Schroedinger operator with a potential
having the symmetry of a honeycomb structure has dispersion surfaces with
conical singularities (Dirac points) at the vertices of its Brillouin zone. No
assumptions are made on the size of the potential. We then prove the robustness
of such conical singularities to a restrictive class of perturbations, which
break the honeycomb lattice symmetry. General small perturbations of potentials
with Dirac points do not have Dirac points; their dispersion surfaces are
smooth. The presence of Dirac points in honeycomb structures is associated with
many novel electronic and optical properties of materials such as graphene.
