We present a full analysis of the spectrum of graphene models on graphs in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for self-similarity by showing that for irrational flux, the spectrum of graphene is a zero measure Cantor set. For arbitrary rational flux, we show the existence of Dirac cones. We also show that for trivial flux, the spectral bands have nontrivial overlap, which leads to the proof of the discrete Bethe-Sommerfeld conjecture for the hexagonal lattice. This talk is based on joint works with S. Becker, J. Fillman and S. Jitomirskaya.
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