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.
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