  One of the cornerstones of modern physics is Einstein's special relativity,
with its constant speed of light and zero photon mass assumptions. Constraint
on the rest mass m_{\gamma} of photons is a fundamental way to test Einstein's
theory, as well as other essential electromagnetic and particle theories. Since
non-zero photon mass can give rise to frequency-(or energy-) dependent
dispersions, measuring the time delay of photons with different frequencies
emitted from explosive astrophysical events is an important and
model-independent method to put such a constraint. The cosmological gamma-ray
bursts (GRBs), with short time scales, high redshifts as well as broadband
prompt and afterglow emissions, provide an ideal testbed for m_{\gamma}
constraints. In this paper we calculate the upper limits of the photon mass
with GRB early time radio afterglow observations as well as multi-band radio
peaks, thus improve the results of Schaefer (1999) by nearly half an order of
magnitude.
