On centers of soluble graphs

Let $G$ be a finite group and $V=\pi(G)$ be a set of all prime divisors of its order. A soluble graph $\Gamma_{sol}(G)$ is a graph with a set of vertices $V$, where two vertices $p$ and $q$ in $V$ are adjacent if there exists a soluble subgroup $H$ of $G$ whose order is divisible by $pq$. We study centers of soluble graphs of finite sporadic and exceptional simple groups of Lie types.