The periodic Schr¨odinger operator H on a discrete periodic graph is treated. The discrete spectrum is estimated for the perturbed operator H±(t) = H±tV, t > 0, where V ≥ 0 is a decaying potential. In the case when the potential has a power asymptotics at infinity, an asymptotics is obtained for the discrete spectrum of the operator H±(t) for a large coupling constant