Automorphism group of the derangement graph (Q640461)

Summary: We prove that the full automorphism group of the derangement graph \(\Gamma_n\) \((n\geq 3)\) is equal to \((R(S_n) \rtimes \text{Inn}(S_n)) \rtimes Z_2\), where \(R(S_n)\) and \(\text{Inn}(S_n)\) are the right regular representation and the inner automorphism group of \(S_n\) respectively, and \(Z_2=\langle\varphi\rangle\) with the mapping \(\varphi:\sigma^\varphi= \sigma^{-1}\), \(\forall\sigma\in S_n\). Moreover, all orbits on the edge set of \(\Gamma_n\) \((n\geq 3)\) are determined.