On the classification of $2$-distance-primitive graphs of small valency

Abstract

A non-complete vertex-transitive graph is called 2-distance-primitive if a vertex stabilizer acts primitively on both the first and second distance neighborhoods of the vertex. Recently, Ruan, Liu and Jin [Discrete Math. 347 (2024), 113978] showed that every connected 2-distance-primitive graph of valency at most 9 either belongs to certain known families or satisfies a collection of further restrictions, with the socle of a vertex stabilizer isomorphic to $\mathrm{PSL}(2,7)$ or $\mathrm{PSL}(2,8)$. In this paper, we resolve these remaining cases and thereby obtain a complete classification of connected 2-distance-primitive graphs of valency at most 9. We also give a short proof of the main result of Ruan, Liu and Jin and identify additional graphs omitted from their list.