The classification of two-distance transitive dihedrants ☆

A vertex transitive graph Gamma is said to be 2-distance transitive if for each vertex u, the group of automorphisms of Gamma fixing the vertex u acts transitively on the set of vertices at distance 1 and 2 from u, while Gamma 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, Malnic and Marusic. 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[b] for some m >= 3 and b >= 2 with mb = 2n. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar