  We investigate the relation between the properties of Brightest Cluster
Galaxies (BCGs) and those of their host clusters. To quantify the properties of
cluster hot gas, we employ the parameter $Z$ of the fundamental plane of X-ray
clusters. It is found that the offset of the BCG from the peak of cluster X-ray
emission is larger for smaller $Z$ clusters. The parameter $Z$ (not the
redshift {\it z}), which mainly depends on virial density $\rho_{\rm {vir}}$,
is considered to represent the formation epoch of a cluster. We thus consider
that the offset of the BCG is correlated with the dynamical equilibrium state
of its host cluster. On the contrary, no significant correlation is found
between the absolute optical magnitude of the BCG and the parameter $Z$. If the
extreme brightness of the BCG is mainly acquired in the course of cluster
evolution by environmental effect, BCGs are expected to be brighter in large
$Z$ clusters. Our result is not consistent with this simplified view. On the
contrary, it is possible that the extreme brightness of the BCG is likely to be
determined in the early history of cluster collapse.
