Integer-Magic Spectra of Sun Graphs

Abstract

Let $\mathbb{N}$ be the set of all positive integers, and ${\mathbb{Z}}_n=\{0,1,2,\dots ,n-1\}$. For any $h\in \mathbb{N}$, a graph $G=(V,E)$ is said to be ${\mathbb{Z}}_h$-magic if there exists a labeling $f:E\to {\mathbb{Z}}_h\setminus \{0\}$ such that the induced vertex labeling $f^{+}:V\to {\mathbb{Z}}_h$, 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.