Critical points of discrete periodic operators

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z 2 - and Z 3 -periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z 2 -periodic graphs.