  Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a
prime $p$ in $\pi(G)$, 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 $\Gamma_A(G)$ of the group $G$, with set of vertices $\pi(G)$,
as follows: Two vertices $p,q\in\pi(G)$ form an edge of $\Gamma_A(G)$ if either
$q\in\pi(A_p(G))$ or $p\in \pi(A_q(G))$. The following result is obtained:
  Theorem: Let $G$ be a finite almost simple group. Then the graph
$\Gamma_A(G)$ 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.
