  We investigate the relationship between brightest cluster galaxies (BCGs) and
their host clusters using a sample of nearby galaxy clusters from the
Representative XMM Cluster Structure Survey (REXCESS). The sample was imaged
with the Southern Observatory for Astrophysical Research (SOAR) in R band to
investigate the mass of the old stellar population. Using a metric radius of
12h^-1 kpc, we found that the BCG luminosity depends weakly on overall cluster
mass as L_BCG \propto M_cl^0.18+-0.07, consistent with previous work. We found
that 90% of the BCGs are located within 0.035 r_500 of the peak of the X-ray
emission, including all of the cool core (CC) clusters. We also found an
unexpected correlation between the BCG metric luminosity and the core gas
density for non-cool core (non-CC) clusters, following a power law of n_e
\propto L_BCG^2.7+-0.4 (where n_e is measured at 0.008 r_500). The correlation
is not easily explained by star formation (which is weak in non-CC clusters) or
overall cluster mass (which is not correlated with core gas density). The trend
persists even when the BCG is not located near the peak of the X-ray emission,
so proximity is not necessary. We suggest that, for non-CC clusters, this
correlation implies that the same process that sets the central entropy of the
cluster gas also determines the central stellar density of the BCG, and that
this underlying physical process is likely to be mergers.
