In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph $\widetilde{G} \rightarrow G=\widetilde{G} /Γ$ with (Abelian) lattice group $Γ$ and periodic magnetic potential $\widetildeβ$. We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on $\widetildeβ$. The magnetic potential may be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and of nanoribbons in the presence of a constant magnetic field.