The Integer-Magic Spectra of Trees

For any positive integer \(h\), a graph \(G=(V,E)\) is said to be \(h\)-magic if there exists a labeling \(l:E(G)→ Z_h -\{0\}\) such that the induced vertex set labeling \(\ l^+ : V(G) → Z_h \) defined by \[ l^+ (v)=∑uv ∈ E(G) \ l(uv) \] is a constant map. The integer-magic spectrum of a graph \(G\), denoted by \(IM(G)\), is the set of all \(h ∈ N\) for which \(G\) is \(h\)-magic. So far, only the integer-magic spectra of trees of diameter at most five have been determined. In this paper, we determine the integer-magic spectra of trees of diameter six and higher.