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We say that G is completely k-magic whenever it is c-sum k-magic for every c ∈ Z k . In this paper, we characterize all completely k-magic regular graphs.

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For positive integer [Formula: see text], a graph [Formula: see text] is said to be [Formula: see text]-magic if the edges of [Formula: see text] can be labeled with the nonzero elements of Abelian group [Formula: see text], where [Formula: see text] (the set of integers) and [Formula: see text] is the group of integers mod [Formula: see text], so that the sum of the labels of the edges incident to any vertex of [Formula: see text] is the same. When this constant sum is [Formula: see text], we say that [Formula: see text] is a zero-sum [Formula: see text]-magic graph. The set of all [Formula: see text] for which [Formula: see text] is a zero-sum [Formula: see text]-magic graph is the null set of [Formula: see text]. In this paper, we will completely determine the null set of the join of a finite number of paths.