  We consider Schr\"odinger operators with periodic magnetic and electric
potentials on periodic discrete graphs. The spectrum of such operators consists
of a finite number of bands. We determine trace formulas for the magnetic
Schr\"odinger 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 magnetic fluxes, electric potentials and
cycles in the quotient graph from some specific cycle sets. Using the trace
formulas we obtain new lower estimates of the total bandwidth for the magnetic
Schr\"odinger operator in terms of geometric parameters of the graph, magnetic
fluxes and electric potentials. We show that these estimates are sharp.
