Some results on the solubility graph of a finite group

Let $G$ be a finite insoluble group with soluble radical $R(G)$. The solubility graph ${\mathrm{\Gamma }}_{\mathrm{S}}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G)$ and two distinct vertices $x$ and $y$ are adjacent if and only if they generate a soluble subgroup of $G$. In this paper, we investigate the several properties of the solubility graph ${\mathrm{\Gamma }}_{\mathrm{S}}(G)$.