  We consider Schr\"odinger operators with periodic electric and magnetic
potentials on periodic discrete graphs. The spectrum of such operators consists
of an absolutely continuous (a.c.) part (a union of a finite number of
non-degenerate bands) and a finite number of eigenvalues of infinite
multiplicity. We prove the following results: 1) the a.c. spectrum of the
magnetic Schr\"odinger operators is empty for specific graphs and magnetic
fields; 2) we obtain necessary and sufficient conditions under which the a.c.
spectrum of the magnetic Schr\"odinger operators is empty; 3) the spectrum of
the magnetic Schr\"odinger operator with each magnetic potential $t\alpha$,
where $t$ is a coupling constant, has an a.c. component for all except finitely
many $t$ from any bounded interval.
