Group magicness of certain planar graphs

Document Type: Research Paper

Authors

1 K.N.Toosi University

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.

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