  A nonzero-mass hypothesis for the photon can produces a frequency-dependent
dispersion of light, which results in arrival-time differences of photons with
different frequencies originating from a given transient source. Extragalactic
fast radio bursts (FRBs), with their low frequency emissions, short time
durations, and long propagation distances, are excellent astrophysical probes
to constrain the rest mass of the photon $m_{\gamma}$. However, the derivation
of a limit on $m_{\gamma}$ is complicated by the similar frequency dependences
of dispersion expected from the plasma and nonzero photon mass effects. If a
handful measurements of redshift for FRBs are available, the different redshift
dependences of the plasma and photon mass contributions to the dispersion
measure (DM) might be able to break dispersion degeneracy in testing the photon
mass. For now, nine FRBs with redshift measurements have been reported, which
can turn this idea into reality. Taking into account the DM contributions from
both the plasma and a possible photon mass, we use the data on the nine FRBs to
derive a combined limit of $m_{\gamma}\leq7.1\times10^{-51}\;{\rm kg}$, or
equivalently $m_{\gamma}\leq4.0\times10^{-15}\; {\rm eV}/c^{2}$ at 68\%
confidence level, which is essentially as good as or represents a factor of 7
improvement over previous limits obtained by the single FRBs. Additionally, a
reasonable estimation for the DM contribution from the host galaxy, $\rm
DM_{host}$, can be simultaneously achieved in our analysis. The rapid progress
in localizing FRBs will further tighten the constraints on both $m_{\gamma}$
and $\rm DM_{host}$.
