[Paper Review] Some other algebraic properties of folded hypercubes

S. Morteza Mirafzal|arXiv (Cornell University)|Mar 22, 2011

Interconnection Networks and Systems5 references3 citations

TL;DR

This paper explicitly constructs the automorphism group of the folded hypercube $FQ_n$ for $n > 3$ as a semidirect product $\mathbb{Z}_2^n \rtimes \text{Sym}(n+1)$, proving that $FQ_n$ is a symmetric graph. The result establishes that $FQ_n$ is both vertex- and edge-transitive with maximum connectivity $n+1$, leveraging linear algebra and group action theory on Cayley graph structures.

ABSTRACT

We construct explicity the automorphism group of the folded hypercube $FQ_n$ of dimension $n>3$, as a semidirect product of $N$ by $M$, where $N$ is isomorphic to the Abelian group $Z_2^n$, and $M$ is isomorphic to $Sym(n+1)$, the symmetric group of degree $n+1$, then we will show that the folded hypercube $FQ_n$ is a symmetric graph.

Motivation & Objective

• To determine the full automorphism group of the folded hypercube $FQ_n$ for $n > 3$.
• To prove that $FQ_n$ is a symmetric graph, extending its known edge transitivity.
• To establish the group structure of $Aut(FQ_n)$ using semidirect product decomposition.
• To show that the stabilizer of a vertex acts transitively on its neighbors, confirming symmetry.
• To demonstrate that $FQ_n$ has maximum connectivity $n+1$ due to its symmetric and regular structure.

Proposed method

• Construct $Aut(FQ_n)$ as a semidirect product $N \rtimes M$, where $N \cong \mathbb{Z}_2^n$ is the group of translations and $M \cong \text{Sym}(n+1)$ is the group of linear automorphisms.
• Define $M$ as the set of linear extensions of bijections from the standard basis $B$ to the generating set $S = B \cup \{u\}$, where $u = \sum e_i$.
• Prove that $N$ is normal in $Aut(FQ_n)$ by showing $f^{-1}gf \in N$ for all $f \in M$, $g \in N$, using conjugation properties of translations.
• Use the fact that $FQ_n$ is a Cayley graph $\Gamma(\mathbb{Z}_2^n, S)$ to analyze automorphisms via group actions on the generating set $S$.
• Show that the stabilizer $G_0$ of the origin acts transitively on $N(0) = S$ by proving $\bar{M} = \text{Sym}(S)$, hence $G_0 \cong \text{Sym}(n+1)$.
• Establish $|Aut(FQ_n)| = 2^n \cdot (n+1)!$ by counting linear extensions and verifying group closure.

Experimental results

Research questions

• RQ1 What is the full automorphism group of the folded hypercube $FQ_n$ for $n > 3$?
• RQ2 Is the folded hypercube $FQ_n$ symmetric, meaning that for any two adjacent vertex pairs, there exists an automorphism mapping one to the other?
• RQ3 How does the automorphism group structure relate to the underlying vector space $\mathbb{Z}_2^n$ and the generating set $S$?
• RQ4 Why does the automorphism group fail to be $\mathbb{Z}_2^n \rtimes \text{Sym}(n+1)$ for $n = 3$?
• RQ5 What is the connectivity of $FQ_n$, and how does symmetry relate to its minimum degree?

Key findings

• The automorphism group of $FQ_n$ for $n > 3$ is isomorphic to $\mathbb{Z}_2^n \rtimes \text{Sym}(n+1)$, with order $2^n \cdot (n+1)!$.
• The folded hypercube $FQ_n$ is a symmetric graph, as the stabilizer of any vertex acts transitively on its neighbors.
• The automorphism group acts transitively on both vertices and edges, confirming vertex and edge transitivity.
• The connectivity of $FQ_n$ is $n+1$, matching its minimum degree, due to its symmetric and edge-transitive structure.
• For $n = 3$, $FQ_3 \cong K_{4,4}$, and the automorphism group has order $2 \cdot (4!)^2 = 1152$, showing the result does not extend to $n = 3$.
• The group $M$ of linear automorphisms is isomorphic to $\text{Sym}(S)$, where $S$ is the generating set of size $n+1$, and acts faithfully on $S$.