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In this article, the lineshape of Fano-like resonance of graphene plasmonic oligomers is investigated as a function of the parameters of the nanostructures, such as disk size, chemical potential and electron momentum relaxation time in mid-infrared frequencies. Also, the mechanism of the optimization is discussed. Furthermore, the environmental index sensing effect of the proposed structure is revealed, and a figure of merit of 25.58 is achieved with the optimized graphene oligomer. The proposed nanostructure could find applications in the fields of chemical or biochemical sensing.