On arc-transitive inner-automorphic Cayley graphs on dihedral groups

TL;DR

This paper characterizes and classifies arc-transitive inner-automorphic Cayley graphs on dihedral groups, providing new examples and a complete classification of certain 2-distance-transitive cases.

Contribution

It identifies four families of such graphs, establishes a necessary condition for others, and completes the classification of all 2-distance-transitive cases.

Findings

Characterized four well-known families of arc-transitive graphs on dihedral groups.

Provided a necessary condition for other such graphs to be inner-automorphic.

Constructed an infinite family of examples satisfying this condition.

Abstract

A Cayley graph $\Cay(G,S)$ is said to be inner-automorphic if $S$ is a union of conjugacy classes of a group $G$, and arc-transitive if its full automorphism group acts transitively on the set of arcs. In this paper, we characterize four well-known families of arc-transitive graphs that arise as connected inner-automorphic Cayley graphs on dihedral groups, and we provide a necessary condition for other connected arc-transitive Cayley graphs on dihedral groups to be inner-automorphic. We further construct an infinite family of examples satisfying this condition, thereby demonstrating the existence of such graphs. Finally, we complete the classification of all 2-distance-transitive connected inner-automorphic Cayley graphs on dihedral groups.