  We consider latin square graphs $\Gamma = \rm{LSG}(H)$ 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 $k = 2$.
