Automorphism groups of Cayley graphs generated by connected transposition sets

Let S be a set of transpositions that generates the symmetric group S n , where n ≥ 3 . The transposition graph T ( S ) is defined to be the graph with vertex set { 1 , … , n } and with vertices i and j being adjacent in T ( S ) whenever ( i , j ) ∈ S . We prove that if the girth of the transposition graph T ( S ) is at least 5, then the automorphism group of the Cayley graph Cay ( S n , S ) is the semidirect product R ( S n ) ⋊ Aut ( S n , S ) , where Aut ( S n , S ) is the set of automorphisms of S n that fixes S . This strengthens a result of Feng on transposition graphs that are trees. We also prove that if the transposition graph T ( S ) is a 4 -cycle, then the set of automorphisms of the Cayley graph Cay ( S 4 , S ) that fixes a vertex and each of its neighbors is isomorphic to the Klein 4-group and hence is nontrivial. We thus identify the existence of 4-cycles in the transposition graph as being an important factor in causing a potentially larger automorphism group of the Cayley graph.