  Understanding the cluster population of clusters of galaxies is of the utmost
importance for using cluster samples in both astrophysical and cosmological
studies. We present an in-depth analysis of the X-ray morphological parameters
of the galaxy clusters and groups detected in the eROSITA Final
Equatorial-Depth Survey (eFEDS). We study the eROSITA X-ray imaging data for a
sample of 325 clusters and groups that are significantly detected in the eFEDS
field. We characterize their dynamical properties by measuring a number of
dynamical estimators: concentration, central density, cuspiness, ellipticity,
power-ratios, photon asymmetry, and Gini coefficient. The galaxy clusters and
groups detected in eFEDS, covering a luminosity range of more than three orders
of magnitude and large redshift range out to 1.2 provide an ideal sample for
studying the redshift and luminosity evolution of the morphological parameters
and characterization of the underlying dynamical state of the sample. Based on
these measurements we construct a new dynamical indicator, relaxation score,
for all the clusters in the sample. We find no evidence for bimodality in the
distribution of morphological parameters of our clusters, rather we observe a
smooth transition from the cool-core to non-cool-core and from relaxed to
disturbed states. A significant evolution in redshift and luminosity is also
observed in the morphological parameters examined in this study after carefully
taking into account the selection effects. We determine that our eFEDS-selected
cluster sample, differently than ROSAT-based cluster samples, is not biased
toward relaxed clusters, but contains a similar fraction of disturbed as SZ
surveys.
