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We investigate the relation between the properties of Brightest ClusterGalaxies (BCGs) and those of their host clusters. To quantify the properties ofcluster hot gas, we employ the parameter $Z$ of the fundamental plane of X-rayclusters. It is found that the offset of the BCG from the peak of cluster X-rayemission is larger for smaller $Z$ clusters. The parameter $Z$ (not theredshift {\it z}), which mainly depends on virial density $\rho_{\rm {vir}}$,is considered to represent the formation epoch of a cluster. We thus considerthat the offset of the BCG is correlated with the dynamical equilibrium stateof its host cluster. On the contrary, no significant correlation is foundbetween the absolute optical magnitude of the BCG and the parameter $Z$. If theextreme brightness of the BCG is mainly acquired in the course of clusterevolution by environmental effect, BCGs are expected to be brighter in large$Z$ clusters. Our result is not consistent with this simplified view. On thecontrary, it is possible that the extreme brightness of the BCG is likely to bedetermined in the early history of cluster collapse.Comment: 18 pages, 9 figures, accepted for publication in Ap

Properties of the Brightest Cluster Galaxy and Its Host Cluster