Automorphism groups of Cayley graphs generated by connected transposition sets

TL;DR

This paper characterizes the automorphism groups of Cayley graphs generated by connected transposition sets, showing they are largely determined by the structure of the transposition graph, with special cases for girth and 4-cycles.

Contribution

It extends previous results by establishing automorphism group structures for transposition graphs with girth at least 5 and identifies 4-cycles as a key factor affecting automorphisms.

Findings

Automorphism group is the semidirect product of $R(S_n)$ and $ ext{Aut}(S_n,S)$ for girth ≥ 5.

Automorphisms fixing a vertex and neighbors form Klein 4-group in 4-cycle case.

Presence of 4-cycles in the transposition graph influences automorphism group size.

Abstract

Let $S$ be a set of transpositions that generates the symmetric group $S_n$, where $n \ge 3$. The transposition graph $T(S)$ is defined to be the graph with vertex set $\{1,\ldots,n\}$ and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in S$. 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 $R(S_n) \rtimes \Aut(S_n,S)$, 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.