Regular set in Cayley sum mgraph
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916275856343040 |
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| author | Seiedali, F. Khosravi, B. Akhlaghi, Z. |
| author_facet | Seiedali, F. Khosravi, B. Akhlaghi, Z. |
| contents | 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular set in Cayley sum mgraph Seiedali, F. Khosravi, B. Akhlaghi, Z. Combinatorics 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 $G$. |
| title | Regular set in Cayley sum mgraph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.03377 |