Stability of vertical and radial graphs in the Euclidean space with density

It is proved that vertical graphs and radial graphs are strongly stable for a certain type of densities in Euclidean space ${\mathbb R}^{n+1}$. Particular cases of these densities include translators, expanders and singular minimal hypersurfaces. Using techniques of calibrations, it is also proved that for densities depending on a spatial coordinate, stationary vertical graphs are weighted minimizers in a certain class of hypersurfaces.