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arXiv 2406.03377math.CO
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Cayley和图中的正则集合

Regular set in Cayley sum mgraph

F. Seiedali, B. Khosravi, Z. Akhlaghi

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AI总结：

本文研究了Cayley和图中的正则集合，证明了有限阿贝尔群子群的正则性条件，并确定了二面体群子群的所有正则对可能情况。

摘要：

图$Γ$的顶点集的一个子集$C$被称为$(α,β)$-正则的，如果$C$导出一个$α$-正则子图，且$C$外的每个顶点恰好与$C$中的$β$个顶点相邻。特别地，如果$C$是有限群$G$的某个带有连接集$S$的Cayley和图中的$(α,β)$-正则集合，则$C$被称为$G$的$(α,β)$-正则集合，而$(0,1)$-正则集合被称为$G$的完美码。用Sq$(G)$和NSq$(G)$分别表示$G$的所有平方元素和非平方元素的集合。作为本文的主要结果之一，我们证明了有限阿贝尔群$G$的子群$H$是$G$的$(α,β)$-正则集合，对于每个$0\leq α\leq |$NSq$(G)\cap H|$且$0\leq β\leq \mathcal{L}(H)$，其中若Sq$(G) \subseteq H$则$\mathcal{L}(H)=|H|$，否则$\mathcal{L}(H)=|$NSq$(G)\cap H|$。作为结果的推论，我们给出了\cite{mama, ma}中主要结果的一个非常简短的证明。此外，我们考虑了二面体群$G=D_{2n}$，对于$G$的每个子群$H$，通过给出适当的连接集$S$，我们确定了$(α, β)$的所有可能情况，其中$H$是$G$的$(α,β)$-正则集合。

英文摘要：

A subset $C$ of the vertex set of a graph $Γ$ is said to be $(α,β)$-regular if $C$ induces an $α$-regular subgraph and every vertex outside $C$ is adjacent to exactly $β$ vertices in $C$. In particular, if $C$ is an $(α,β)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(α,β)$-regular set of $G$ and a $(0,1)$-regular set is called a perfect code of $G$. By Sq$(G)$ and NSq$(G)$ we mean the set of all square elements and non-square elements of $G$. As one of the main results in this note, we show that a subgroup $H$ of a finite abelian group $G$ is an $(α,β)$-regular set of $G$, for each $0\leq α\leq |$NSq$(G)\cap H|$ and $0\leq β\leq \mathcal{L}(H)$, where $\mathcal{L}(H)=|H|$, if Sq$(G) \subseteq H$ and $\mathcal{L}(H)=|$NSq$(G)\cap H|$, otherwise. As a consequence of our result we give a very brief proof for the main results in \cite{mama, ma}. Also, we consider the dihedral group $G=D_{2n} $ and for each subgroup $H $ of $G$, by giving an appropriate connection set $S$, we determine each possibility for $(α, β)$, where $H$ is an $(α,β)$-regular set of $G$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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