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Khuluq, M., Krisnawati, V., Hidayat, N.. (1404). On $\mathbb{Z}_k-$vertex-magic labeling of prime graph $PG(\mathbb{Z}_n)$. فیزیولوژی و بیوتکنولوژی آبزیان, 13(2), 27-37. doi: 10.22124/jart.2024.26022.1601
M. H. Khuluq; V. H. Krisnawati; N. Hidayat. "On $\mathbb{Z}_k-$vertex-magic labeling of prime graph $PG(\mathbb{Z}_n)$". فیزیولوژی و بیوتکنولوژی آبزیان, 13, 2, 1404, 27-37. doi: 10.22124/jart.2024.26022.1601
Khuluq, M., Krisnawati, V., Hidayat, N.. (1404). 'On $\mathbb{Z}_k-$vertex-magic labeling of prime graph $PG(\mathbb{Z}_n)$', فیزیولوژی و بیوتکنولوژی آبزیان, 13(2), pp. 27-37. doi: 10.22124/jart.2024.26022.1601
Khuluq, M., Krisnawati, V., Hidayat, N.. On $\mathbb{Z}_k-$vertex-magic labeling of prime graph $PG(\mathbb{Z}_n)$. فیزیولوژی و بیوتکنولوژی آبزیان, 1404; 13(2): 27-37. doi: 10.22124/jart.2024.26022.1601

On $\mathbb{Z}_k-$vertex-magic labeling of prime graph $PG(\mathbb{Z}_n)$

Journal of Algebra and Related Topics
دوره 13، شماره 2، اسفند 2025، صفحه 27-37    اصل مقاله (164.54 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22124/jart.2024.26022.1601
نویسندگان
M. H. Khuluq* 1؛ V. H. Krisnawati2؛ N. Hidayat3
1Department of Mathematics, Faculty of Mathematics and Sciences, University of Brawijaya, Malang, Indonesia
2Department of Mathematics, Faculty of Mathematics and Sciences, University of Brawijaya, Malang, Indonesia
3Department of Mathematics, Faculty of Mathematics and Sciences, University of Brawijaya, Malang, Indonesia
چکیده
Let $G=(V(G),E(G))$ be a graph, $(\mathcal{A},+)$ be an Abelian group with identity $0_{\mathcal{A}}$, and $(\mathcal{{R}},+,\cdot)$ be a ring. The $\mathcal{A}$-vertex-magic labeling of $G$ is a mapping from $V(G)$ to $\mathcal{A}-\{0_{\mathcal{A}}\}$ such that the total labels of every adjacent vertex with $u$ are equal for every $u$ in $V(G)$. The prime graph over ${\mathcal{R}}$, denoted by $PG(\mathcal{R})$, is a graph with $V(PG(\mathcal{R}))={\mathcal{R}}$ such that $uv$ is an edge if and only if $u\mathcal{R}v=\{0_{\mathcal{R}}\}$ or $v{\mathcal{R}}u=\{0_{\mathcal{R}}\}$, for every vertex $u\neq v$. In this article, we discuss the $\mathbb{Z}_k$-vertex-magic labeling of the prime graph over ther ring $\mathbb{Z}_n$. We study some literature to develop the properties of $\mathbb{Z}_k$-vertex-magic labeling of $PG(\mathcal{R})$. We investigate some classes of prime graphs over ring $\mathbb{Z}_n$ for $n=p, n=p^2,$ and $n=pq$, with $p\neq q$ primes.
کلیدواژه‌ها
Abelian group؛ Group-vertex-magic labeling؛ Prime graph؛ Ring
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