Finite 2-distance transitive graphs

A non-complete graph $Γ$ is said to be $(G,2)$-distance transitive if $G$ is a subgroup of the automorphism group of $Γ$ that is transitive on the vertex set of $Γ$, and for any vertex $u$ of $Γ$, the stabilizer $G_u$ is transitive on the sets of vertices at distance 1 and 2 from $u$. This paper investigates the family of $(G,2)$-distance transitive graphs that are not $(G,2)$-arc transitive. Our main result is the classification of such graphs of valency not greater than 5.

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