  For a massive quantum fermionic field, we investigate the vacuum expectation
values (VEVs) of the charge and current densities induced by an external
magnetic flux in a two-dimensional circular ring. Both the irreducible
representations of the Clifford algebra are considered. On the ring edges the
bag (infinite mass) boundary conditions are imposed for the field operator.
This leads to the Casimir type effect on the vacuum characteristics. The radial
current vanishes. The charge and the azimuthal current are decomposed into the
boundary-free and boundary-induced contributions. Both these contributions are
odd periodic functions of the magnetic flux with the period equal to the flux
quantum. An important feature that distinguishes the VEVs of the charge and
current densities from the VEV of the energy density, is their finiteness on
the ring edges. The current density is equal to the charge density for the
outer edge and has the opposite sign on the inner edge. The VEVs are peaked
near the inner edge and, as functions of the field mass, exhibit quite
different features for two inequivalent representations of the Clifford
algebra. We show that, unlike the VEVs in the boundary-free geometry, the
vacuum charge and the current in the ring are continuous functions of the
magnetic flux and vanish for half-odd integer values of the flux in units of
the flux quantum. Combining the results for two irreducible representations, we
also investigate the induced charge and current in parity and time-reversal
symmetric models. The corresponding results are applied to graphene rings with
the electronic subsystem described in terms of the effective Dirac theory with
the energy gap. If the energy gaps for two valleys of the graphene hexagonal
lattice are the same, the charge densities corresponding to the separate
valleys cancel each other, whereas the azimuthal current is doubled.
