Recent studies in different types of galaxies reveal that the product of the central density and the core radius ( \rho _ { c } r _ { c } ) is a constant .
However , some empirical studies involving galaxy clusters suggest that the product \rho _ { c } r _ { c } depends weakly on the total dark halo mass .
In this article , we re-analyse the hot gas data from 106 clusters and obtain a surprisingly tight scaling relation : \rho _ { c } \propto r _ { c } ^ { -1.46 \pm 0.16 } .
This result generally agrees with the claims that \rho _ { c } r _ { c } is not a constant for all scales of structure .
Moreover , this relation does not support the velocity-dependent cross section of dark matter if the core formation is due to the self-interaction of dark matter .