Automorphism groups of Cayley graphs generated by general transposition sets

TL;DR

This work analyzes Cayley graphs $\mathrm{Cay}(S_n,T)$ generated by transpositions, linking their automorphism groups to the corresponding transposition graph $G(T)$. It proves that for $n \ge 5$, if $G(T) \ncong K_n$, the graph is normal and $\mathrm{Aut}(\mathrm{Cay}(S_n,T)) = R(S_n) \rtimes \mathrm{Aut}(S_n,T)$, yielding $\mathrm{Aut}(\mathrm{Cay}(S_n,T)) \cong S_n \times \mathrm{Aut}(G(T))$. The result confirms Ganesan's conjecture by showing non-normality arises only in the exceptional cases $G(T) \cong C_4$ or $K_n$, and provides a complete classification for $n \ge 5$. The approach leverages Whitney-type connections between automorphisms of the transposition graph and the Cayley graph via line graphs, highlighting the role of $G(T)$ in determining symmetry properties relevant to interconnection networks and sorting-inspired Cayley structures.

Abstract

In this paper we study the Cayley graph $\mathrm{Cay}(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group is a direct product of $S_n$ and the automorphism group of the transposition graph associated to $T$. This provides an affirmative answer to a conjecture raised by Ganesan in arXiv:1703.08109, showing that $\mathrm{Cay}(S_n,T)$ is normal if and only if the transposition graph is not $C_4$ or $K_n$.

Figures (1)

• Figure 1: Subgraph of $G$ induced by vertices $1, 2, 3$ and any $k \notin \{1,2,3\}$. Existing and non-existing edges are denoted by solid and dotted lines, respectively.

Theorems & Definitions (18)

• Theorem 1
• Corollary 2
• Theorem 3: Whitney1932
• Theorem 4: ganesan2015automorphism
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof