Structure of the automorphism group of the augmented cube graph

Abstract

\noindent The augmented cube graph $A{Q}_n$ is the Cayley graph of ${\mathbb{Z}}_2^n$ with respect to the set of $2n-1$ generators $\{e_1,e_2,\dots ,e_n,00\dots 0011,00\dots 0111,11\dots 1111\}$. It is known that the order of the automorphism group of the graph $A{Q}_n$ is $2^{n+3}$, for all $n\ge 4$. In the present paper, we obtain the structure of the automorphism group of $A{Q}_n$ 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 $A{Q}_3$ is non-normal and that $A{Q}_n$ is normal for all $n\ge 4$. We also analyze the clique structure of $A{Q}_4$ and show that the automorphism group of $A{Q}_4$ is isomorphic to that of $A{Q}_3$: \[ \Aut(AQ_4) \cong \Aut(AQ_3) \cong (D_8 \times D_8) \rtimes C_2.\] All the nontrivial blocks of $A{Q}_4$ are also determined.

References in corpus (2)

• Automorphism group of the complete transposition graph
• Some other algebraic properties of folded hypercubes