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

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematics and Sciences, University of Brawijaya, Malang, Indonesia

Abstract

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.

Keywords

Journal of Algebra and Related Topics

Articles in Press, Accepted Manuscript
Available Online from 04 December 2024

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