  Umbral dots (UDs) are transient, bright features observed in the umbral
region of a sunspot. We study the physical properties of UDs observed in
sunspots of different sizes. The aim of our study is to relate the physical
properties of umbral dots with the large-scale properties of sunspots. For this
purpose, we analyze high-resolution G-band images of 42 sunspots observed by
{\em Hinode}/SOT, located close to disk center. The images were corrected for
instrumental stray-light and restored with the modeled PSF. An automated
multi-level tracking algorithm was employed to identify the UDs located in
selected G-band images. Furthermore, we employed HMI/SDO, limb-darkening
corrected, full disk continuum images to estimate the sunspot phase and epoch
for the selected sunspots. The number of UDs identified in different umbrae
exhibits a linear relation with the umbral size. The observed filling factor
ranges from 3\% to 7\% and increases with the mean umbral intensity. Moreover,
the filling factor shows a decreasing trend with the umbral size. We also found
that the observed mean and maximum intensities of UDs are correlated with the
mean umbral intensity. However, we do not find any significant relationship
between the mean (and maximum) intensity and effective diameter of umbral dots
with the sunspot area, epoch, and decay rate. We suggest that this lack of
relation could either be due to the distinct transition of spatial scales
associated with overturning convection in the umbra or the shallow depth
associated with umbral dots, or both the above.
