On the Saxl graphs of primitive groups with soluble stabilisers

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