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We consider normalized Laplacians and their perturbations by periodicpotentials (Schr\"odinger operators) on periodic discrete graphs. The spectrumof the operators consists of an absolutely continuous part (a union of a finitenumber of non-degenerate bands) and a finite number of flat bands, i.e.,eigenvalues of infinite multiplicity. We obtain estimates of the Lebesguemeasure of the spectrum in terms of geometric parameters of the graphs and showthat they become identities for some class of graphs. We determine two-sidedestimates on the length of the first spectral band and on the effective mass atthe bottom of the spectrum of the Laplace and Schr\"odinger operators. Inparticular, these estimates yield that the first spectral band of Schr\"odingeroperators is non-degenerate.

Spectral estimates for Schr\"odinger operators on periodic discrete graphs