  We calculate the total galactic Shapiro delay to the Crab pulsar by including
the contributions from the dark matter as well as baryonic matter along the
line of sight. The total delay due to dark matter potential is about 3.4 days.
For baryonic matter, we included the contributions from both the bulge and the
disk, which are approximately 0.12 and 0.32 days respectively. The total delay
from all the matter distribution is therefore 3.84 days. We also calculate the
limit on violations of Einstein's equivalence principle by using observations
of "nano-shot" giant pulses from the Crab pulsar with time-delay $<0.4$~ns as
well as using time differences between radio and optical photons observed from
this pulsar. Using the former, we obtain a limit on violation of Einstein's
equivalence principle in terms of the PPN parameter $\Delta \gamma < 2.41\times
10^{-15}$. From the time-difference between simultaneous optical and radio
observations, we get $\Delta \gamma < 1.54\times 10^{-9}$. We also point out
differences in our calculation of Shapiro delay and that from two recent papers
(arXiv:1612.00717 and arXiv:1608.07657), which used the same observations to
obtain a corresponding limit on $\Delta \gamma$.
