  Strong evidence for the Helling-Downs correlation curves have been reported
by multiple pulsar timing array (PTA) collaborations in middle 2023. In this
work, we investigate the graviton mass bounds by analyzing the observational
data of the overlap ruduction 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, $m_{g}\lesssim8.6\times10^{-24}\mathrm{eV}$ from NANOGrav and
$m_{g}\lesssim3.8\times10^{-23}\mathrm{eV}$ from CPTA. We also study the
cosmic-variance limit on the graviton mass bounds, i.e.,
$\sigma_{m_{g}}^{\mathrm{CV}}=4.8\times10^{-24}\mathrm{eV}\times
f/(10~\mathrm{year})^{-1}$, 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.
