Generic potentials on any connected periodic graph produce no flat bands.
Periodic and limit-periodic discrete Schr\"odinger operators
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The theory of discrete periodic and limit-periodic Schr\"odinger operators is developed. In particular, the Floquet--Bloch decomposition is discussed. Furthermore, it is shown that an arbitrarily small potential can add a gap for even periods. In dimension two, it is shown that for coprime periods small potential terms don't add gaps thus proving a Bethe--Sommerfeld type statement. Furthermore limit-periodic potentials whose spectrum is an interval are constructed.
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Absence of flat bands for discrete periodic graph operators with generic potentials
Generic potentials on any connected periodic graph produce no flat bands.