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.
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