Generic potentials on any connected periodic graph produce no flat bands.
Rare Flat Bands for Periodic Graph Operators
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As a corollary of our main results, we prove that for any connected $\mathbb{Z}^d$-periodic graph, when edge weights and potentials are treated as variables, the corresponding periodic graph operators generically (i.e., outside a proper algebraic subset of the variable space) do not have flat bands.
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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.