  Let $G$ be a transitive permutation group on a finite set $\Omega$ and recall
that a base for $G$ is a subset of $\Omega$ with trivial pointwise stabiliser.
The base size of $G$, denoted $b(G)$, is the minimal size of a base. If
$b(G)=2$ then we can study the Saxl graph $\Sigma(G)$ of $G$, which has vertex
set $\Omega$ and two vertices are adjacent if they form a base. This is a
vertex-transitive graph, which is conjectured to be connected with diameter at
most $2$ when $G$ is primitive. In this paper, we combine probabilistic and
computational methods to prove a strong form of this conjecture for all almost
simple primitive groups with soluble point stabilisers. In this setting, we
also establish best possible lower bounds on the clique and independence
numbers of $\Sigma(G)$ and we determine the groups with a unique regular
suborbit, which can be interpreted in terms of the valency of $\Sigma(G)$.
