Distance Two Vertex-Magic Graphs

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 \( l^*: V(G) \to A \) defined by

\[l^*(v) = \sum_{c \in E(v)} l(e)\]

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 \( \Delta M(G) \). In this paper, the distance two magic spectra of certain classes of graphs will be determined.