  Using extensive tight-binding calculations, we investigate (including the
spin) the Aharonov-Bohm (AB) effect in monolayer and bilayer trigonal and
hexagonal graphene rings with zigzag boundary conditions. Unlike the previous
literature, we demonstrate the universality of integer (hc/e) and half-integer
(hc/2e) values for the period of the AB oscillations as a function of the
magnetic flux, in consonance with the case of mesoscopic metal rings. Odd-even
(in the number of Dirac electrons, N) sawtooth-type patterns relating to the
halving of the period have also been found; they are more numerous for a
monolayer hexagonal ring, compared to the cases of a trigonal and a bilayer
hexagonal ring. Additional more complicated patterns are also present,
depending on the shape of the graphene ring. Overall, the AB patterns repeat
themselves as a function of N with periods proportional to the number of the
sides of the rings.

Mesoscale and Nanoscale Physics;Materials Science;Atomic and Molecular Clusters