Some results on the main supergraph of finite groups

Let $G$ be a finite group. The main supergraph $\mathcal{S}(G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o(x) \mid o(y)$ or $o(y)\mid o(x)$. In this paper, we will show that $G\cong \mathrm{PSL}(2,p)$ or $\mathrm{PGL}(2,p)$ if and only if $\mathcal{S}(G)\cong \mathcal{S}(\mathrm{PSL}(2,p))$ or $\mathcal{S}(\mathrm{PGL}(2,p))$, respectively. Also, we will show that if $M$ is a sporadic simple group, then $G\cong M$ if only if $\mathcal{S}(G)\cong \mathcal{S}(M)$.