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

• Authors: Yan-Quan Feng
• Venue: Journal of Combinatorial Theory Series B
• Year: 2006

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 {ij|(i j) ∈ T}. In this paper it is shown that if n ≥ 3, then the automorphism group of the transposition graph Tra(T) is isomorphic to Aut(Sn, T) = {α ∈ Aut(Sn)|Tα = T} 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.