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Abstract
A Cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts transitively on edges. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. This is a partial answer to the question of Praeger (1999).
Highlights
Let G be a finite group, and let S be a subset of G such that 1G ∈/ S
The Cayley graph X Cay G, S of G on S is defined as the graph with a vertex set V X G and edge set E X {{g, sg} | g ∈ G, s ∈ S}
A part of Aut X may be described in terms of automorphisms of G, that is, the normalizer NAut X G G Aut G, S, a semidirect product of G by Aut G, S, where Aut G, S {σ ∈ Aut G | Sσ S}
Summary
Let G be a finite group, and let S be a subset of G such that 1G ∈/ S. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. A Cayley graph X Cay G, S is said to be vertex transitive, edge transitive, and arc transitive if its automorphism group Aut X is transitive on the vertex set V X , edge set E X , and arc set A X , respectively. A subgroup of the automorphism group of a graph X is said to be s-regular if it acts regularly on the set of s-arcs of X.
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