Integer-Magic Spectra of Trees of Diameter Five

For any \( h \in \mathbb{Z} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by

\[
l^+(v) = \sum_{uv \in E(G)} l(uv)
\]

is a constant map. For a given graph \( G \), the set of all \( h \in \mathbb{Z}_+ \) for which \( G \) is \( h \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will determine the integer-magic spectra of trees of diameter five.