We have systematically studied the single-particle states in quantum rings produced by a set of concentric circular gates over a graphene sheet placed on a substrate. The resulting potential profiles and the interaction between the graphene layer and the substrate are considered within the Dirac Hamiltonian in the framework of the envelope function approximation. Our simulations allow microscopic mapping of the character of the electron and hole quasi-particle solutions according to the applied voltage. General conditions to control and operate the bound state solutions are described as functions of external and controllable parameters that will determine the optical properties ranging from metallic to semiconductor phases. Contrasting behaviors are obtained when comparing the results for repulsive and attractive voltages as well as for variation of the relative strength of the graphene-substrate coupling parameter. © 2012 IOP Publishing Ltd.