  We study the energy spectra and wavefunctions of graphene rings formed from
metallic armchair ribbons, near zero energy, to search for properties which may
be identified with "effective broken time reversal symmetry" (EBTRS).
Appropriately chosen corner junctions are shown to impose phase shifts in the
wavefunctions that at low energies have the same effect as effective flux tubes
passing near the ribbon surface. Closing the ribbon into a ring captures this
flux and yields properties that may be understood as signatures of EBTRS. These
include a gap in the spectrum around zero energy, which can be removed by the
application of real magnetic flux through the ring. Spectra of five and seven
sided rings are also examined, and it is shown these do not have particle-hole
symmetry, which may also be understood as a consequence of EBTRS, and is
connected to the curvature induced in the system when such rings are formed.
Effects of deviations from the ideal geometries on the spectra are also
examined.
