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