Comparing classes of highly symmetric graphs: From $2$-arc-transitive to $2$-distance-transitive

Abstract

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first- and second-step neighborhoods. This concept simultaneously generalizes both distance-transitive graphs and $2$-arc-transitive graphs. In this paper, we first determine the vertex-quasiprimitive types of $2$-distance-transitive graphs of odd order, partially answering a question posed by A. Devillers, M. Giudici, C. H. Li and C. E. Praeger in 2012. We then prove that a $2$-distance-transitive graph of valency $p+1$, where $p$ is a prime, is $2$-arc-transitive if and only if it has girth at least $4$. We also show that every locally-primitive $2$-distance-transitive graph of valency at most $8$ is $2$-arc-transitive, with the icosahedron as the unique exception. Finally, we prove that if $\Gamma$ is a $G$-locally-primitive, $(G,2)$-distance-transitive graph of valency at least $3$ and $G$ is soluble, then either $Γ\cong \K_{p,p}$ for some prime $p$, or the order of $\Gamma$ is not square-free.