$Z_k$-Magic Labeling of Some Families of Graphs

Document Type: Research Paper

Authors

1 Principal and Head of the Research Centre,Department of Mathematics,Govindammal Aditanar College for Women,Tiruchendur,Tamilnadu,INDIA

2 Department of Mathematics Holy Cross College, Nagercoil, Tamilnadu, India.

Abstract

For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-\textit{magic}  if there exists a labeling $f:E(G) \rightarrow A-\{0\}$ such that, the vertex labeling $f^+$  defined as $f^+(v) = \sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-\textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$  the group of integers modulo $k$. These $Z_k$-magic graphs are referred to as $k$-\textit{magic} graphs. In this paper we prove that the total graph, flower graph,  generalized prism graph, closed helm graph, lotus inside a circle graph, $G\odot\overline{K_m}$, $m$-splitting graph of a path and  $m$-shadow graph of a path are $Z_k$-magic graphs.

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