On the coincidence of Grünberg–Kegel graphs of a finite simple group and its proper subgroup

Let $G$ be a finite group. The spectrum of $G$ is the set $\omega(G)$ of orders of its elements. The subset of prime elements of $\omega(G)$ is denoted by $\pi(G)$. The spectrum $\omega(G)$ of a group $G$ defines its prime graph (or Grünberg–Kegel graph) $\Gamma(G)$ with vertex set $\pi(G)$, in which any two different vertices $r$ and $s$ are adjacent if and only if the number $rs$ belongs to the set $\omega(G)$. We describe all the cases when the prime graphs of a finite simple group and of its proper subgroup coincide.