On the automorphism groups of us-Cayley graphs

Abstract

Let G be a finite abelian group written additively with identity 0, and Ω be an inverse closed generating subset of G such that 0 ∉ Ω. We say that Ω has the property ‘‘us’’ (unique summation), whenever for every 0 ≠ g ∈ G if there are s₁, s₂, s₃, s₄ ∈ Ω such that s₁ + s₂ = g = s₃ + s₄, then we have {s₁, s₂} = {s₃, s₄}. We say that a Cayley graph Γ = Cay(G;Ω) is a us-Cayley graph, whenever G is an abelian group and the generating subset Ω has the property ‘‘us’’. In this paper, we show that if Γ = Cay(G;Ω) is a us-Cayley graph, then Aut(Γ) = L(G) ⋊ A, where L(G) is the left regular representation of G and A is the group of all automorphism groups θ of the group G such that θ(Ω) = Ω. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including Möbius ladders and k-ary n-cubes.