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On continuous spectrum of magnetic Schrödinger operators on periodic discrete graphs

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