Automorphism groups of Cayley graphs generated by connected transposition sets

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Let S be a set of transpositions that generates the symmetric group Sₙ, 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 (Sₙ,S) is the semidirect product R(Sₙ) (Sₙ,S), where (Sₙ,S) is the set of automorphisms of Sₙ 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 (S₄,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.