Some results on the solubility graph of a finite group

Abstract

Let $G$ be a finite insoluble group with soluble radical $R(G)$. The solubility graph ${\Gamma }_{\mathrm{S}}(G)$ 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 ${\Gamma }_{\mathrm{S}}(G)$.

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