In General Relativity (GR), the graviton is massless. However, a common feature in several theoretical alternatives of GR is a non-zero mass for the graviton. These theories can be described as massive gravity theories. Despite many theoretical complexities in these theories, on phenomenological grounds, the implications of massive gravity have been widely used to put bounds on graviton mass. One of the generic implications of giving a mass to the graviton is that the gravitational potential will follow a Yukawa-like fall off. We use this feature of massive gravity theories to probe the mass of graviton by using the largest gravitationally bound objects, namely galaxy clusters. In this work, we use the mass estimates of galaxy clusters measured at various cosmologically defined radial distances measured via weak lensing (WL) and Sunyaev-Zeldovich (SZ) effect. We also use the model independent values of Hubble parameter $H(z)$ smoothed by a non-parametric method, Gaussian process. Within $1sigma$ confidence region, we obtain the mass of graviton $m_g < 5.9 times 10^{-30}$ eV with the corresponding Compton length scale $lambda_g > 6.82$ Mpc from weak lensing and $m_g < 8.31 times 10^{-30}$ eV with $lambda_g > 5.012$ Mpc from SZ effect. This analysis improves the upper bound on graviton mass obtained earlier from galaxy clusters.
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