Automorphisms of Cayley graphs generated by transposition sets

TL;DR

This paper characterizes the automorphism groups of Cayley graphs generated by transposition sets with girth at least 5, generalizing previous results and applying to interconnection network topologies.

Contribution

It provides a general formula for automorphism groups of such Cayley graphs, extending prior work and connecting to line graph automorphisms.

Findings

Automorphism group is the semidirect product of the right regular representation and automorphisms fixing S.

When components are isomorphic, automorphisms correspond to line graph automorphisms.

Results include automorphism groups of extended cube graphs used in network topologies.

Abstract

Let $S$ be a set of transpositions such that the girth of the transposition graph of $S$ is at least 5. It is shown that the automorphism group of the Cayley graph of the permutation group $H$ generated by $S$ is the semidirect product $R(H) \rtimes \Aut(H,S)$, where $R(H)$ is the right regular representation of $H$ and $\Aut(H,S)$ is the set of automorphisms of $H$ that fixes $S$ setwise. Furthermore, if the connected components of the transposition graph of $S$ are isomorphic to each other, then $\Aut(H,S)$ is isomorphic to the automorphism group of the line graph of the transposition graph of $S$. This result is a common generalization of previous results by Feng, Ganesan, Harary, Mirafzal, and Zhang and Huang. As another special case, we obtain the automorphism group of the extended cube graph that was proposed as a topology for interconnection networks.