Strong evidence for the Helling-Downs correlation curves have been reported by multiple pulsar timing array (PTA) collaborations in the middle of 2023. In this work, we investigate the graviton mass bounds by analyzing the observational data of the overlap reduction functions from the NANOGrav 15-year data release and CPTA first data release. The results from our data analysis display the state-of-the-art upper limits on the graviton mass at 90% confidence level, namely, $m_g\lesssim 8.6\times {10}^{-24}\mathrm{eV}$ from NANOGrav and $m_g\lesssim 3.8\times {10}^{-23}\mathrm{eV}$ from CPTA. We also study the cosmic-variance limit on the graviton mass bounds, i.e., ${\sigma }_{m_g}^{\mathrm{CV}}=4.8\times {10}^{-24}\mathrm{eV}\times f/(10\mathrm{year}{)}^{-1}$, with $f$ being a typical frequency band of PTA observations. This is equivalent to the cosmic-variance limit on the speed of gravitational waves, i.e., ${\sigma }_{v_g}^{\mathrm{CV}}=0.07c$, with $c$ being the speed of light. Moreover, we discuss potential implications of these results for scenarios of ultralight tensor dark matter.

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• Astrophysical studies of gravity
• Gravitational waves