The Inter-magic Spectra of Trees

Abstract

For any positive integer $h$, 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. The integer-magic spectrum of a graph $G$, denoted by $IM(G)$, is the set of all $h\in \mathbb{N}$ 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.

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