Distance Two Vertex-Magic Graphs

Abstract

Given an abelian group $A$, a graph $G=(V,E)$ is said to have a distance two magic labeling in $A$ if there exists a labeling $l:E(G)\to A–\{0\}$ such that the induced vertex labeling $l^{\ast }:V(G)\to A$ defined by

$$l^{\ast }(v)=\sum _{c\in E(v)}l(e)$$

is a constant map, where $E(v)=\{e\in E(G):d(v,e)<2\}$. The set of all $h\in {\mathbb{Z}}_{+}$, for which $G$ has a distance two magic labeling in ${\mathbb{Z}}_h$, is called the distance two magic spectrum of $G$ and is denoted by $\mathrm{\Delta }M(G)$. In this paper, the distance two magic spectra of certain classes of graphs will be determined.