A graphene-based band-stop filter structure for the surface plasmonic polariton in the mid-infrared wave band is proposed, which consists of a periodic array of single-layered graphene. The numerical simulation results show that the slight variation of the chemical potential of the graphene nanoribbon results in the shift of resonance wavelength. This structure possesses a sensitivity of 1100 nm/RIU and a figure of merit of 138 in the mid-infrared spectrum range, which can be used as a highly sensitive refractive index sensor.