Quotient-complete arc-transitive latin square graphs from groups

Abstract

We consider latin square graphs $\Gamma =\mathrm{LSG}(\mathrm{H})$ of the Cayley table of a given finite group $H$. We characterize all pairs $(\Gamma ,G)$, where $G$ is a subgroup of autoparatopisms of the Cayley table of $H$ such that $G$ acts arc-transitively on $\Gamma$ and all nontrivial $G$-normal quotient graphs of $\Gamma$ are complete. We show that $H$ must be elementary abelian and determine the number $k$ of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with $k=1$ or $k=2$.