Cayley graph on symmetric groups with generating block transposition sets

TL;DR

This paper studies the automorphism groups of Cayley graphs on symmetric groups generated by block transpositions, revealing their structure and properties relevant to bioinformatics and graph theory.

Contribution

It characterizes the automorphism group of these Cayley graphs and explores their subgraphs, introducing new structural insights and conjectures.

Findings

Automorphism group of Cayley graph is a product of right translation and a dihedral group.

Subgraph on $S_n$ is a $2(n-2)$-regular graph with automorphism group $ extsf{D}_{n+1}.

Subgraph of maximum cliques is 3-regular, Hamiltonian, and vertex-transitive.

Abstract

This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $\Gamma$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose automorphism group is $\textsf{D}_{n+1}$. Furthermore, $\Gamma$ has as many as $n+1$ maximum cliques of size $2.$ Also, its subgraph $\Gamma(V)$ whose vertices are those in these cliques is a $3$-regular, Hamiltonian, and vertex-transitive graph.