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$k$-MAGIC VERTEX LABELING WITH GROUP ELEMENTS
DOI:
https://doi.org/10.17654/0974165825028Abstract
Let $A$ be a finite abelian group. Let $G$ be simple graph with $|A|$ vertices. By labeling of vertex set $V(G)$, we mean a one-to-one function $\quad \phi: V(G) \rightarrow A$. A weight function is a function $w_i: \phi(V(G)) \rightarrow A$, where $i=1,2$. In this article, we consider the following two weight functions:
$$
w_1(\phi(v))=\prod_{u v \in E(G)} \phi(u)
$$
and
$$
w_2(\phi(v))=\phi(v) \cdot \prod_{u v \in E(G)} \phi(u) .
$$
The graph $G$ is said to be ( $A, w_i, k$ ) graph if the cardinality of the set $\left\{w_i(\phi(v)): \phi(v) \in \phi(V(G))\right\}$ is $k$. The results of this paper ascertain some well-known classes of graphs to be $\left(A, w_i, k\right)$ for specific instances of $A$ and $k$.
Received: April 6, 2025
Revised: April 14, 2025
Accepted: May 16, 2025
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