Abstract

Context. The first Spectrum-Roentgen-Gamma (SRG) eROSITA public release contains 12 247 clusters and groups from its first 6 months of operation. We used the offset between the brightest cluster galaxy (BCG) and the X-ray peak (DBCG - X) to classify the cluster dynamical state of 3946 galaxy clusters and groups. Aims. We aim to investigate the evolution of the merger and relaxed cluster distributions with redshift and mass, and the distributions' impact on the BCG. Methods. We used the X-ray peak from the eROSITA survey and the BCG position from the LS DR10 optical data, which includes the DECam eROSITA Survey optical data, to measure the DBCG - X offset. We modelled the distribution of DBCG - X, in units of R-500, as the sum of two Rayleigh distributions representing the cluster's relaxed and disturbed populations, and explored their evolution with redshift and mass. To explore the impact of the cluster's dynamical state on the BCG luminosity, we separated the main sample according to the dynamical state. We defined clusters as relaxed if DBCG - X < 0.25R(500), disturbed if DBCG - X > 0.5R(500), and 'diverse' if 0.25R(500) < DBCG - X < 0.5R(500). Results. We find no evolution of the merging fraction with redshift or mass. The width of the relaxed distribution increases with redshift, while the width of the two Rayleigh distributions decreases with mass. The analysis reveals that BCGs in relaxed clusters are brighter than BCGs in both the disturbed and diverse cluster populations. The most significant differences are found for high-mass clusters at higher redshifts. Conclusions. The results suggest that BCGs in low-mass clusters are less centrally bound than those in high-mass systems, irrespective of the dynamical state. Over time, BCGs in relaxed clusters progressively align with the potential centre. This alignment correlates with their luminosity growth relative to BCGs in dynamically disturbed clusters, underscoring the critical role of the cluster's dynamical state in regulating BCG evolution.