Abstract
AbstractA Cayley graphΓ\Gammaon a groupGis called adual Cayley graphonGif the left regular representation ofGis a subgroup of the automorphism group ofΓ\Gamma(note that the right regular representation ofGis always an automorphism group ofΓ\Gamma). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible. A few problems worth further research are also proposed.
Highlights
Graphs considered in this article are finite and undirected
For a graph Γ, we denote by VΓ and AutΓ its vertex set and its full automorphism group, respectively
For a vertex α of Γ, we denote by Γ (α) ≔ {β ∈ VΓ|β ∼ α} the neighbor set of α in Γ, where β ∼ α means that β is adjacent to α
Summary
Graphs considered in this article are finite and undirected. For a graph Γ , we denote by VΓ and AutΓ its vertex set and its full automorphism group, respectively. Cayley graphs were introduced by Cayley [1] in 1878, which provide an important and rich source of transitive graphs, stated as follows: a graph Γ is called a Cayley graph on a group G if there is a subset S ⊆ G\ {1}, with S = S−1 ≔ {g−1|g ∈ S}, such that VΓ = G and two vertices g and h are adjacent if and only if hg−1 ∈ S This Cayley graph is denoted by Cay(G, S). The aim of this article is to investigate the identification, construction and transitivity of dual Cayley graphs and propose several problems needing further research
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