On the Sylow graph of a group and Sylow normalizers

Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a prime $p$ in $π(G)$, the set of all prime divisors of the order of $G$. The automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define the Sylow graph $Γ_A(G)$ of the group $G$, with set of vertices $π(G)$, as follows: Two vertices $p,q\inπ(G)$ form an edge of $Γ_A(G)$ if either $q\inπ(A_p(G))$ or $p\in π(A_q(G))$. The following result is obtained: Theorem: Let $G$ be a finite almost simple group. Then the graph $Γ_A(G)$ is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.