  The solubility graph $\Gamma_S(G)$ associated with a finite group $G$ is a
simple graph whose vertices are the elements of $G$, and there is an edge
between two distinct vertices if and only if they generate a soluble subgroup.
In this paper, we focus on the set of neighbors of a vertex $x$ which we call
the solubilizer of $x$ in $G$, $\mathrm{Sol}_G(x)$, investigating both
arithmetic and structural properties of this set.
