Velocity Dispersions of Brightest Cluster Galaxies and Their Host
  Clusters
  We explore connections between brightest cluster galaxies (BCGs) and their
host clusters. We first construct a HeCS-omnibus cluster sample including 227
galaxy clusters within $0.02 < z < 0.30$; the total number of spectroscopic
members from MMT/Hectospec and SDSS observations is 52325. Taking advantage of
the large spectroscopic sample, we compute physical properties of the clusters
including the dynamical mass and cluster velocity dispersion ($\sigma_{cl}$).
We also measure the central stellar velocity dispersion of the BCGs
($\sigma_{*,BCGs}$) to examine the relation between BCG velocity dispersion and
cluster velocity dispersion for the first time. The observed relation between
BCG velocity dispersion and the cluster velocity dispersion is remarkably
tight. Interestingly, the $\sigma_{*, BCG} / \sigma_{cl}$ ratio decreases as a
function of $\sigma_{cl}$ unlike the prediction from the numerical simulation
of Dolag et al. (2010). The trend in $\sigma_{*, BCG} / \sigma_{cl}$ suggests
that the BCG formation is more efficient in lower mass halos.
