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

Abstract

Let T be a set of transpositions of the symmetric group S n . 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 ( S n , T ) = { α ∈ Aut ( S n ) | T α = T } and if T is a minimal generating set of S n then the automorphism group of the Cayley graph Cay ( S n , T ) is the semiproduct R ( S n ) ⋊ Aut ( S n , T ) , where R ( S n ) is the right regular representation of S n . 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 S n : if T is a minimal generating set of S n and the automorphism group of the transposition graph Tra ( T ) is trivial then the automorphism group of the Cayley graph Cay ( S n , T ) is isomorphic to S n .