The classification of two-distance transitive dihedrants

Abstract

A vertex transitive graph Γ is said to be 2-distance transitive if for each vertex u, the group of automorphisms of Γ fixing the vertex u acts transitively on the set of vertices at distance 1 and 2 from u, while Γ is said to be 2-arc transitive if its automorphism group is transitive on the set of 2-arcs. Then 2-arc transitive graphs are 2-distance transitive. In 2008, the 2-arc transitive Cayley graphs on dihedral groups were classified by Du, Malnič and Marušič. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order 2n is either 2-arc transitive, or isomorphic to the complete multipartite graph $K_{m\left[b\right]}$ for some $m\ge 3$ and $b\ge 2$ with $mb=2n$.