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We consider latin square graphs \(\varGamma = \text {LSG}(H)\) of the Cayley table of a given finite group H. We characterize all pairs \((\varGamma ,G)\), where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on \(\varGamma \) and all nontrivial G-normal quotient graphs of \(\varGamma \) 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 \(k = 2\).

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups