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| Paper IPM / M / 18003 |
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| Abstract: | |
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A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(\alpha,\beta)$-regular if $C$ induces an $\alpha$-regular subgraph and every vertex outside $C$ is adjacent to exactly $\beta$ vertices in $C$. In particular, if $C$ is an $(\alpha,\beta)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(\alpha,\beta)$-regular set 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 $(\alpha,\beta)$-regular set of $G$, for each $0\leq \alpha \leq |$NSq$(G)\cap H|$ and $0\leq \beta \leq \mathcal{L}(H)$, where $\mathcal{L}(H)=|H|$, if Sq$(G) \subseteq H$ and $\mathcal{L}(H)=|$NSq$(G)\cap H|$, otherwise. As a consequence we easily get that $H$ is a $(0,1)$-regular, if and only if either Sq$(G)\subseteq H$ or NSq$(G)\cap H\not=\emptyset$. The proof of this result is given by X. Ma, K. Wang, and Y. Yang in 2022, in a longer method. Also, X. Ma, M. Feng, and K. Wang in 2020, gave a sufficient and necessary condition for a subgroup $H$ to be a $(0,1)$-regular subgroup of an abelian group $G$. Our new result makes that result much easier to gain.
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 $(\alpha, \beta)$, where $H$ is an $(\alpha,\beta)$-regular set of $G$.
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