Cubic Graphical Regular Representations of $\mathrm{PSU}_3(q)$

TL;DR

This paper proves that the projective special unitary group $ ext{PSU}_3(q)$ admits a cubic graphical regular representation if and only if $q \\geq 4$, and constructs such representations for these cases.

Contribution

It establishes the existence of cubic GRRs for $ ext{PSU}_3(q)$ when $q \\geq 4$ and provides explicit constructions, advancing understanding of GRRs for finite simple groups.

Findings

$ ext{PSU}_3(q)$ has a cubic GRR if and only if $q \\geq 4$

Explicit cubic GRRs are constructed for all such $q$

Supports the conjecture that only finitely many simple groups lack cubic GRRs

Abstract

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that $\mathrm{PSU}_3(q)$ has a cubic GRR if and only if $q\geq4$. Moreover, a cubic GRR of $\mathrm{PSU}_3(q)$ is constructed for each of these $q$.