  We analyze theoretically the electronic properties of Aharonov-Bohm rings
made of graphene. We show that the combined effect of the ring confinement and
applied magnetic flux offers a controllable way to lift the orbital degeneracy
originating from the two valleys, even in the absence of intervalley
scattering. The phenomenon has observable consequences on the persistent
current circulating around the closed graphene ring, as well as on the ring
conductance. We explicitly confirm this prediction analytically for a circular
ring with a smooth boundary modelled by a space-dependent mass term in the
Dirac equation. This model describes rings with zero or weak intervalley
scattering so that the valley isospin is a good quantum number. The tunable
breaking of the valley degeneracy by the flux allows for the controlled
manipulation of valley isospins. We compare our analytical model to another
type of ring with strong intervalley scattering. For the latter case, we study
a ring of hexagonal form with lattice-terminated zigzag edges numerically. We
find for the hexagonal ring that the orbital degeneracy can still be controlled
via the flux, similar to the ring with the mass confinement.
