  We present a study of relations between the masses of the central
supermassive black holes (SMBHs) and the atmospheric gas temperatures and
luminosities measured within a range of radii between $R_{\rm e}$ and 5$R_{\rm
e}$, for a sample of 47 early-type galaxies observed by the {\it Chandra X-ray
Observatory}. We report the discovery of a tight correlation between the
atmospheric temperatures of the brightest cluster/group galaxies (BCGs) and
their central SMBH masses. Furthermore, our hydrostatic analysis reveals an
approximately linear correlation between the total masses of BCGs ($M_{\rm
tot}$) and their central SMBH masses ($M_{\rm BH}$). State-of-the-art
cosmological simulations show that the SMBH mass could be determined by the
binding energy of the halo through radiative feedback during the rapid black
hole growth by accretion, while for the most massive galaxies mergers are the
chief channel of growth. In the scenario of a simultaneous growth of central
SMBHs and their host galaxies through mergers, the observed linear correlation
could be a natural consequence of the central limit theorem.
