In this work a graphene-photonic crystal composite structure is proposed, in which graphene nanoribbons are placed on top of a photonic crystal that is composed of two alternating dielectric materials. The optical properties of the composite structure are investigated in the infrared regime, by varying the Fermi energy, the layer number of the graphene, as well as the polarization angle of the incident light. Plasmonic resonance effects are revealed and the resonance wavelength, the peak width, and the peak intensity are determined. Quantitatively, the polarization-angle dependent peak intensity is well fitted to a trigonometric function. It is demonstrated that the resonance peak is quite sensitive to the dielectric material that is adjacent to the graphene. It is also shown that the structure has a high sensitivity of 7660 nm/RIU to the refractive index of the surrounding environment, with a figure of merit of 6.38. Based on the results, the graphene-photonic crystal composite structure proposed in this work may have potential applications in designing of plasmonic devices, such as ones that can be used to detect subtle variations in the refractive index of surrounding environments.