Automorphism group of the derangement graph

Abstract

In this paper, we prove that the full automorphism group of the derangement graph Γn (n ≥ 3) is equal to (R(Sn) ⋊ Inn(Sn)) ⋊ Z2, where R(Sn) and Inn(Sn) are the right regular representation and the inner automorphism group of Sn respectively, and Z2 = 〈φ〉 with the mapping φ: σφ = σ-1, ∀ σ ∈ Sn. Moreover, all orbits on the edge set of Γn (n ≥ 3) are determined.