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We support, with new fitting instruments and the analysis of more recentexperimental data, the proposal of a relationship between the mass of aSupermassive Black Hole (SMBH) and the kinetic energy of random motions in thehost elliptical galaxy. The first results obtained in a previous paper with 13elliptical galaxies are now confirmed by the new data and an enlarged sample.We find $M_{BH} \propto (M_{G} \sigma^{2}/c^2)^\beta$ with $0.8 \leq \beta \leq1$ depending on the different fitting methods and samples used. The meaningfulcase $\beta = 1$ is carefully analyzed. Furthermore, we test the robustness ofour relationship including in the sample also lenticular and spiral galaxiesand we show that the result does not change. Finally we find a strongercorrelation between the mass of the galaxy and the corresponding velocitydispersion that allows to connect our relationship to the $M_{BH} \propto\sigma^\alpha$ law. With respect to this law, our relationship has theadvantage to have a smaller scatter.Comment: 17 pages, 3 figures, accepted for publication in IJMP

IMPROVED TESTS ON THE RELATIONSHIP BETWEEN THE KINETIC ENERGY OF GALAXIES AND THE MASS OF THEIR CENTRAL BLACK HOLES