  We demonstrate that hexagonal graphene nanoflakes with zigzag edges display
quantum interference (QI) patterns analogous to benzene molecular junctions. In
contrast with graphene sheets, these nanoflakes also host magnetism. The
cooperative effect of QI and magnetism enables spin-dependent quantum
interference effects that result in a nearly complete spin polarization of the
current, and holds a huge potential for spintronic applications. We understand
the origin of QI in terms of symmetry arguments, which show the robustness and
generality of the effect. This also allows us to devise a concrete protocol for
the electrostatic control of the spin polarization of the current by breaking
the sublattice symmetry of graphene, by deposition on hexagonal boron nitride,
paving the way to switchable spin-filters. Such a system benefits of all the
extraordinary conduction properties of graphene, and at the same time, it does
not require any external magnetic field to select the spin polarization, as
magnetism emerges spontaneously at the edges of the nanoflake.
