On Group-Magic Trees, Double Trees and Abbreviated Double Trees

For \( k > 0 \), we call a graph \( G = (V,E) \) \( k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k^* \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \), defined by

\[
l^+(v) = \sum\{l(u,v): (u,v) \in E(G)\}
\]

is a constant map. We denote the set of all \( k \) such that \( G \) is \( k \)-magic by \(\text{IM}(G)\). We call this set the \textbf{\emph{integer-magic spectrum}} of \( G \). We investigate these sets for trees, double trees, and abbreviated double trees. We define group-magic spectrum for \( G \) similarly. Finally, we show that a tree is \( k \)-magic, \( k > 2 \), if and only if it is \( k \)-label reducible.