In this paper, the minimax design problem of graph filter using Chebyshev polynomial approximation (CPA) is studied. First, conventional CPA graph filter design is investigated to show that it does not provide the minimax design, so the peak error of the spectral response of the designed filter is not minimized. Then, a spectral transformation is used to convert the minimax design problem of CPA graph filter into the one of the type-I linear-phase FIR digital filter such that the Parks-McClellan method can be employed to obtain the optimal filter coefficients of minimax design. Next, using the recurrence relation of Chebyshev polynomials, an implementation structure of CPA graph filter with low computational complexity is presented. Finally, a graph signal denoising application example is illustrated to show the usefulness of the proposed CPA graph filter.