  In this article, we relate the spectrum of the discrete magnetic Laplacian
(DML) on a finite simple graph with two structural properties of the graph: the
existence of a perfect matching and the existence of a Hamiltonian cycle of the
underlying graph. In particular, we give a family of spectral obstructions
parametrised by the magnetic potential for the graph to be matchable (i.e.,
having a perfect matching) or for the existence of a Hamiltonian cycle. We base
our analysis on a special case of the spectral preorder introduced in [FCLP20a]
and we use the magnetic potential as a spectral control parameter.
