Automorphism groups of a class of cubic Cayley graphs on symmetric groups

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Let Sₙ denote the symmetric group of degree n with n≥ 3. Set S={cₙ=(12 n),cₙ⁻¹,(12)}. Let Γₙ=Cay(Sₙ,S) be the Cayley graph on Sₙ with respect to S. In this paper, we show that Γₙ (n≥ 13) is a normal Cayley graph, and that the full automorphism group of Γₙ is equal to Aut(Γₙ)=R(Sₙ) (ϕ) Sₙ Z₂, where R(Sₙ) is the right regular representation of Sₙ, ϕ=(12)(3n)(4n-1)(5n-2) (∈ Sₙ), and Inn(ϕ) is the inner isomorphism of Sₙ induced by ϕ.