^{2}Department of Mathematics, Karaj Branch, Islamic Azad Uneversity, Karaj-Iran

Abstract

Let $A$ be a non-trivial abelian group and $A^{*}=A\setminus \{0\}$. A graph $G$ is said to be $A$-magic graph if there exists a labeling $l:E(G)\rightarrow A^{*}$ such that the induced vertex labeling $l^{+}:V(G)\rightarrow A$, define by $$l^+(v)=\sum_{uv\in E(G)} l(uv)$$ is a constant map. The set of all constant integers such that $\sum_{u\in N(v)} l(uv)=c$, for each $v\in N(v)$, where $N(v)$ denotes the set of adjacent vertices to vertex $v$ in $G$, is called the index set of $G$ and denoted by ${\rm In}_{A}(G).$
In this paper we determine the index set of certain planar graphs for $\mathbb{Z}_{h}$, where $h\in \mathbb{N}$, such as wheels and fans.