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The S\'ersic $R^{1/n}$ model is the best approximation known to date fordescribing the light distribution of stellar spheroidal and disk components,with the S\'ersic index $n$ providing a direct measure of the central radialconcentration of stars. The S\'ersic index of a galaxy's spheroidal component,$n_{sph}$, has been shown to tightly correlate with the mass of the centralsupermassive black hole, $M_{BH}$. The $M_{BH}-n_{sph}$ correlation is alsoexpected from other two well known scaling relations involving the spheroidluminosity, $L_{sph}$: the $L_{sph}-n_{sph}$ and the $M_{BH}-L_{sph}$.Obtaining an accurate estimate of the spheroid S\'ersic index requires acareful modelling of a galaxy's light distribution and some studies have failedto recover a statistically significant $M_{BH}-n_{sph}$ correlation. With theaim of re-investigating the $M_{BH}-n_{sph}$ and other black hole mass scalingrelations, we performed a detailed (i.e.~bulge, disks, bars, spiral arms,rings, halo, nucleus, etc.) decomposition of 66 galaxies, with directlymeasured black hole masses, that had been imaged at $3.6\rm~\mu m$ withSpitzer. In this paper, the third of this series, we present an analysis of the$L_{sph}-n_{sph}$ and $M_{BH}-n_{sph}$ diagrams. While early-type(elliptical+lenticular) and late-type (spiral) galaxies split into two separaterelations in the $L_{sph}-n_{sph}$ and $M_{BH}-L_{sph}$ diagrams, they reuniteinto a single $M_{BH} \propto n_{sph}^{3.39 \pm 0.15}$ sequence with relativelysmall intrinsic scatter ($\epsilon \simeq 0.25 \rm~dex$). The black hole massappears to be closely related to the spheroid central concentration of stars,which mirrors the inner gradient of the spheroid gravitational potential.

Supermassive black holes and their host spheroids III. The $M_{BH} - n_{sph}$ correlation