Integer-Magic Spectra of Trees with Diameters at Most Four

For any \( k \in \mathbb{N} \), a graph \( G = (V,E) \) is said to be \( \mathbb{Z}_k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \) defined by
\[
l^+(v) = \sum_{u \in N(v)} l(uv)
\]
is a constant map. For a given graph \( G \), the set of all \( k \in \mathbb{Z}_+ \) for which \( G \) is \( \mathbb{Z}_k \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will consider trees whose diameters are at most \( 4 \) and will determine their integer-magic spectra.