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CAYLEY GRAPHS AND GRAPHICAL REGULAR REPRESENTATIONS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 15 September 2023, pp. 161-162
- Print publication:
- February 2024
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Finite transitive groups having many suborbits of cardinality at most 2 and an application to the enumeration of Cayley graphs
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 30 January 2023, pp. 345-366
- Print publication:
- February 2024
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- Article
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Let G be a finite transitive group on a set
$\Omega $, let 
$\alpha \in \Omega $, and let 
$G_{\alpha }$ be the stabilizer of the point 
$\alpha $ in G. In this paper, we are interested in the proportion 
$$ \begin{align*} \frac{|\{\omega\in \Omega\mid \omega \textrm{ lies in a }G_{\alpha}\textrm{-orbit of cardinality at most 2}\}|}{|\Omega|}, \end{align*} $$that is, the proportion of elements of
$\Omega $ lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than 
$5/6$, then each element of 
$\Omega $ lies in a suborbit of cardinality at most 2, and hence G is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound 
$5/6$.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.
