On arc-transitive inner-automorphic Cayley graphs on dihedral groups
Jun-Jie Huang, Jin-Hua Xie

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 is said to be inner-automorphic if is a union of conjugacy classes of a group , 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.
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