Integer-Magic Spectra of Sun Graphs

Abstract

Let \(\mathbb{N}\) be the set of all positive integers, and \(\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\). For any \(h \in \mathbb{N}\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_h\)-magic if there exists a labeling \(f: E \rightarrow \mathbb{Z}_h \setminus \{0\}\) such that the induced vertex labeling \(f^+: V \rightarrow \mathbb{Z}_h\), defined by \(f^+(v) = \sum_{uv \in E(v)} f(uv)\), is a constant map. The integer-magic spectrum of \(G\) is the set \(\text{JM}(G) = \{h \in \mathbb{N} \mid G \text{ is } \mathbb{Z}_h\text{-magic}\}\). A sun graph is obtained from attaching a path to each pair of adjacent vertices in an \(n\)-cycle. In this paper, we show that the integer-magic spectra of sun graphs are completely determined.