We consider Schrodinger 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 Schrodinger 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 Schrodinger operators is empty; 3) the spectrum of the magnetic Schrodinger operator with each magnetic potential $talpha$, where $t$ is a coupling constant, has an a.c. component for all except finitely many $t$ from any bounded interval.
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