We want to study how the velocity segregation and the radial profile of the velocity dispersion depend on the prominence of the brightest cluster galaxies (BCGs). We divide a sample of 102 clusters and groups of galaxies into four bins of magnitude gap between the two brightest cluster members. We then compute the velocity segregation in bins of absolute and relative magnitudes. Moreover, for each bin of magnitude gap we compute the radial profile of the velocity dispersion. When using absolute magnitudes, the segregation in velocity is limited to the two brightest bins and no significant difference is found for different magnitude gaps. However, when we use relative magnitudes, a trend appears in the brightest bin: the larger the magnitude gap, the larger the velocity segregation. We also show that this trend is mainly due to the presence, in the brightest bin, of satellite galaxies in systems with small magnitude gaps: in fact, if we study separately central galaxies and satellites, this trend is mitigated and central galaxies are more segregated than satellites for any magnitude gap. A similar result is found in the radial velocity dispersion profiles: a trend is visible in central regions (where the BCGs dominate) but, if we analyse the profile using satellites alone, the trend disappears. In the latter case, the shape of the velocity dispersion profile in the centre of systems with different magnitude gaps show three types of behaviours: systems with the smallest magnitude gaps have an almost flat profile from the centre to the external regions; systems with the largest magnitude gaps show a monothonical growth from the low values of the central part to the flat ones in the external regions; finally, systems with $1.0 < Delta m_{12} le 1.5$ show a profile that peaks in the centres and then decreases towards the external regions. We suggest that two mechanisms could be respons....
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