On the prime graph of simple groups

Let $G$ be a finite group, let $\pi (G)$ be the set of prime divisors of $\mathrm{\mid }G\mathrm{\mid }$ and let $\mathrm{\Gamma }(G)$ be the prime graph of $G$. This graph has vertex set $\pi (G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that $\mathrm{\Gamma }(G)=\mathrm{\Gamma }(H)$.