Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets

Let T be a set of transpositions of the symmetric group Sn. The transposition graph Tra(T) of T is the graph with vertex set {1,2,…,n} and edge set . In this paper it is shown that if n⩾3, then the automorphism group of the transposition graph Tra(T) is isomorphic to and if T is a minimal generating set of Sn then the automorphism group of the Cayley graph Cay(Sn,T) is the semiproduct R(Sn)⋊Aut(Sn,T), where R(Sn) is the right regular representation of Sn. As a result, we generalize a theorem of Godsil and Royle [C.D. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001, p. 53] regarding the automorphism groups of Cayley graphs on Sn: if T is a minimal generating set of Sn and the automorphism group of the transposition graph Tra(T) is trivial then the automorphism group of the Cayley graph Cay(Sn,T) is isomorphic to Sn.