An updated comparison of the $M_{bullet}$ vs $M_{G}sigma^2$ relation with $M_{bullet}$ vs $sigma$ and the problem of the masses of galaxies

We have studied, in a series of papers, the properties of the $M_{bullet}$ versus $M_{G}sigma^2$ relation and we have found that it is useful to describe the evolution of galaxies in the same way as the HR diagram does for stars and to predict the masses of Supermassive Black Holes that are difficult to be guessed using other scaling relations. In this paper, analyzing five samples of galaxies, we find that this relation has intrinsic scatter similar to the $M_{bullet} - sigma$, but follows the theoretical models much better than the $M_{bullet} - sigma$. Furthermore, we analyze the role of the bulge mass in the behavior of $M_{bullet}$ versus $M_{G}sigma^2$ relation because the difference with the $M_{bullet} - sigma$ is often determined by the choice of the right sample of galactic masses.