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) Zₕ -\0\\) such that the induced vertex set labeling \ (\ l^+: V (G) Zₕ \) defined by \ l^+ (v) =ₔₕ ₄ (₆) \ 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.