[Paper Review] On the Automorphism Groups of Regular Hyper-Stars and Folded Hyper-Stars

S. Morteza Mirafzal|arXiv (Cornell University)|Mar 18, 2011

graph theory and CDMA systems6 references3 citations

TL;DR

This paper determines the automorphism groups of regular hyper-star graphs $HS(2k,k)$ and folded hyper-star graphs $FHS(2k,k)$, proving that only $HS(4,2)$ and $FHS(4,2)$ are Cayley graphs. It establishes that for $k \geq 3$, these graphs are not Cayley graphs due to structural constraints on regular subgroups within their automorphism groups.

ABSTRACT

The hyper-star graph $HS(n,k)$ is defined as follows : its vertex-set is the set of $ {0,1} $-sequences of length $n$ with weight $k$, where the weight of a sequence $v$ is the number of $1^,s$ in $v$, and two vertices are adjacent if and only if one can be obtained from the other by exchanging the first symbol with a different symbol (1 with 0, or 0 with 1) in another position. In this paper, we will find the automorphism groups of regular hyper-star and folded hyper-star graphs. Then, we will show that, only the graphs HS(4,2) and FHS(4,2) are Cayley graphs.

Motivation & Objective

• To determine the automorphism groups of regular hyper-star graphs $HS(2k,k)$ and folded hyper-star graphs $FHS(2k,k)$.
• To investigate whether these graphs are Cayley graphs, particularly for $k \geq 3$.
• To analyze the action of automorphism groups on vertex sets and their stabilizers to determine regularity and transitivity.
• To establish structural constraints on potential regular subgroups within $Aut(HS(2k,k))$ that would be required for a Cayley graph structure.
• To resolve the case distinction between $k=2$ and $k \geq 3$ in the context of Cayley graph realization.

Proposed method

• Uses graph isomorphism between $HS(n,k)$ and the Johnson graph $S(n,k)$ to analyze symmetries via set systems and characteristic functions.
• Applies group action theory: studies the action of $Aut(\Gamma)$ on vertex and edge sets, focusing on transitivity and stabilizer subgroups.
• Employs the concept of $m$-homogeneity to analyze permutation group actions on subsets, particularly on $Y = \{2, \dots, 2k\}$.
• Analyzes the structure of regular subgroups $R \leq Aut(HS(2k,k))$ by decomposing elements as $\tilde{\sigma}\theta^i$, where $\theta$ is a complementation map.
• Uses combinatorial divisibility arguments: derives $k-1 \mid k+1$ from $\binom{2k-1}{k-2} \mid \frac{1}{2}\binom{2k}{k}$, restricting $k \in \{1,2,3\}$.
• Performs explicit cycle analysis in $Sym(6)$ for $k=3$, showing that any involution fixing 1 must fix a vertex, contradicting regularity.

Experimental results

Research questions

• RQ1 What is the automorphism group of the regular hyper-star graph $HS(2k,k)$?
• RQ2 What is the automorphism group of the folded hyper-star graph $FHS(2k,k)$?
• RQ3 For which values of $k$ is $HS(2k,k)$ or $FHS(2k,k)$ a Cayley graph?
• RQ4 Why do $HS(4,2)$ and $FHS(4,2)$ qualify as Cayley graphs while others do not?
• RQ5 What structural constraints prevent $Aut(HS(2k,k))$ from containing a regular subgroup for $k \geq 3$?

Key findings

• The automorphism group of $HS(2k,k)$ and $FHS(2k,k)$ is isomorphic to $H = NQ$, where $N$ and $Q$ are specific subgroups of permutations and complementations.
• $HS(4,2)$ is isomorphic to $C_6$, the 6-cycle, and its automorphism group is the dihedral group $D_{12}$ of order 12.
• $FHS(4,2)$ is isomorphic to $K_{3,3}$, the complete bipartite graph, with automorphism group of order 72.
• For $k \geq 3$, neither $HS(2k,k)$ nor $FHS(2k,k)$ is a Cayley graph, as no regular subgroup exists in their automorphism group.
• The only values of $k$ for which $HS(2k,k)$ is a Cayley graph are $k=2$, where it is $C_6$, and $k=3$, but $k=3$ is ruled out by cycle structure contradiction.
• For $k=3$, the assumption that $R \leq Aut(HS(6,3))$ acts regularly leads to a contradiction because any involution fixing 1 must fix a vertex, violating semiregularity.