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In this paper, we propose a high-sensitivity optical sensor at terahertz frequencies based on a composite structure containing a one-dimensional photonic crystal (1D PC) coated with a layer of monolayer graphene. Between the 1D PC and the graphene there is a sensing medium. This high-sensitivity phenomenon originates from the excitation of optical resonance between the graphene and the 1D PC. The proposed sensor is highly sensitive to the Fermi energy of graphene, the thickness and refractive index of the sensing medium, and the number of graphene layers. By selecting appropriate parameters, the maximum sensitivity (<inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mn>407.36</mn> </mrow> <mo>∘</mo> </msup> <mo>/</mo> <mi>RIU</mi> </mrow> </semantics> </math> </inline-formula>) is obtained. We believe the proposed configuration is promising for fabricating graphene-based biosensor- or gas-sensor devices and other related applications in the terahertz band.