Finite transitive groups having many suborbits of cardinality at most two and an application to the enumeration of Cayley graphs

Let $G$ be a finite transitive group on a set $\mathrm{\Omega }$, let $\alpha\in \Omega$andlet$G_\alpha$bethestabilizerofthepoint$\alpha$in$G$. In this paper, we are interested in the proportion $$\frac{\mathrm{\mid }\{\omega \in \mathrm{\Omega }\mid \omega \text{ lies in a }{G}_{\alpha }\text{-orbit of cardinality at most two}\}\mathrm{\mid }}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }},$$ that is, the proportion of elements of $\mathrm{\Omega }$ lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than $5\mathrm{/}6$, then each element of $\mathrm{\Omega }$ lies in a suborbit of cardinality at most two and hence $G$ is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound $5\mathrm{/}6$. We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group $G$ containing a regular subgroup $R$, we determine an upper bound on the number of Cayley graphs on $R$ containing $G$ in their automorphism groups.