Automorphism group of the modified bubble-sort graph

TL;DR

This paper determines the automorphism group of the modified bubble-sort graph of dimension n, showing it is isomorphic to the direct product of the symmetric group and a dihedral group for all n ≥ 5.

Contribution

It provides a complete structural description of the automorphism group of the modified bubble-sort graph, extending to normal Cayley graphs generated by transpositions.

Findings

Automorphism group is S_n × D_{2n} for n ≥ 5

Complete structural characterization of the automorphism group

Extension to normal Cayley graphs generated by transpositions

Abstract

The modified bubble-sort graph of dimension $n$ is the Cayley graph of $S_n$ generated by $n$ cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension $n$ is $S_n \times D_{2n}$, for all $n \ge 5$. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct product decomposition is seen to hold for arbitrary normal Cayley graphs generated by transposition sets.