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We prove the discrete Bethe-Sommerfeld conjecture on the graphene lattice, on its dual lattice (the triangular lattice), and on the extended Harper lattice. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criteria on the periods that ensure that no gaps open, and we characterize those energies at which gaps may open in the perturbative regime. In all three cases, we provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero.

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Journal d'Analyse Mathematique

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