▎ 摘　　要

In this paper, the optical bistability (OB) and optical multistability (OM) properties of weak probe light in a defect dielectric medium doped by four-level graphene nanostructure is theoretically discussed. The double dark resonance can arise by linear polarized control laser fields which consist of linear left and right circularly polarized light. We show that by adjusting the Rabi-frequencies of control fields, frequency detuning of bichromatic electric fields, the intensity threshold and hysteric curves of OB and OM can be manipulated. Moreover, the thickness of the slab is considered as a controllable parameter which can impact the OB and OM behaviors of weak probe light in a defect slab. We find that the transition from OB to OM or vice versa can be possible by adjusting the thickness of the slab. Our results may provide some new application on Nano-scale devices in future all-optical communication and quantum information technologies.