Structure of the automorphism group of the augmented cube graph

TL;DR

This paper determines the detailed structure of the automorphism group of augmented cube graphs, revealing their symmetry properties and normality for different dimensions, and analyzes their clique structure.

Contribution

It explicitly characterizes the automorphism group of $AQ_n$ as a semidirect product, and shows the normality of these graphs for all dimensions $n eq 3$, including detailed analysis for $AQ_4$.

Findings

Automorphism group of $AQ_n$ is isomorphic to $Z_2^n times D_8$ for $n eq 3$.

$AQ_3$ is non-normal, while $AQ_n$ is normal for all $n eq 3$.

Automorphism groups of $AQ_3$ and $AQ_4$ are explicitly described.

Abstract

\noindent The augmented cube graph $AQ_n$ is the Cayley graph of $\mathbb{Z}_2^n$ with respect to the set of $2n-1$ generators $\{e_1,e_2, \ldots,e_n, 00\ldots0011, 00\ldots0111, 11\ldots1111 \}$. It is known that the order of the automorphism group of the graph $AQ_n$ is $2^{n+3}$, for all $n \ge 4$. In the present paper, we obtain the structure of the automorphism group of $AQ_n$ to be \[ \Aut(AQ_n) \cong \mathbb{Z}_2^n \rtimes D_8~~(n \ge 4),\] where $D_8$ is the dihedral group of order 8. It is shown that the Cayley graph $AQ_3$ is non-normal and that $AQ_n$ is normal for all $n \ge 4$. We also analyze the clique structure of $AQ_4$ and show that the automorphism group of $AQ_4$ is isomorphic to that of $AQ_3$: \[ \Aut(AQ_4) \cong \Aut(AQ_3) \cong (D_8 \times D_8) \rtimes C_2.\] All the nontrivial blocks of $AQ_4$ are also determined.