Quotient-complete arc-transitive latin square graphs from groups

TL;DR

This paper characterizes certain highly symmetric latin square graphs derived from finite groups, identifying conditions under which they have arc-transitive automorphism groups with specific quotient properties, leading to new examples of diameter two graphs.

Contribution

It provides a complete characterization of groups and automorphism subgroups acting arc-transitively on latin square graphs with complete normal quotients, revealing new infinite families.

Findings

H must be elementary abelian

Determined the number of complete normal quotients k

Constructed new diameter two arc-transitive graphs

Abstract

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 $k = 2$.