  We consider Schr\"odinger operators with periodic potentials on periodic
discrete graphs. Their spectrum consists of a finite number of bands. We
determine trace formulas for the Schr\"odinger operators. The proof is based on
the decomposition of the Schr\"odinger operators into a direct integral and a
specific representation of fiber operators. The traces of the fiber operators
are expressed as finite Fourier series of the quasimomentum. The coefficients
of the Fourier series are given in terms of the potentials and cycles in the
quotient graph from some specific cycle sets. We also present the trace
formulas for the heat kernel and the resolvent of the Schr\"odinger operators
and the determinant formulas.
