  We consider reduced-dimensionality models of honeycomb lattices in magnetic
fields and report results about the spectrum, the density of states,
self-similarity, and metal/insulator transitions under disorder. We perform a
spectral analysis by which we discover a fractal Cantor spectrum for irrational
magnetic flux through a honeycomb, prove the existence of zero energy Dirac
cones for each rational flux, obtain an explicit expansion of the density of
states near the conical points, and show the existence of mobility edges under
Anderson-type disorder. Our results give a precise description of de Haas-van
Alphen and Quantum Hall effects, and provide quantitative estimates on
transport properties. In particular, our findings explain experimentally
observed asymmetry phenomena by going beyond the perfect cone approximation.
