On the solubilizer of an element in a finite group

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.