A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $vin V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph $G$. The idea is illustrated in detail by dealing with three regular graphs. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings.
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