Normalized Laplacians and their perturbations by periodic potentials (Schrödinger operators) on periodic discrete graphs are treated. The spectrum of such an operator consists of an absolutely continuous part, which is the union of a finite number of nondegenerate bands, and a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. Estimates for the Lebesgue measure of the spectrum in terms of geometric parameters of the graphs are obtained and it is shown that these estimates become identities for some graphs. Two-sided estimates are given for the lengths of the first spectral bands and for the effective masses at the bottom of the spectrum for the Laplace and Schrödinger operators. In particular, these estimates show that the first spectral band of the Schrödinger operators is nondegenerate.