The null set of the join of paths

For positive integer k, a graph G is said to be k-magic if the edges of G can be labeled with the nonzero elements of Abelian group ℤ k, where ℤ 1= ℤ (the set of integers) and ℤ k is the group of integers mod k≥ 2, so that the sum of the labels of the edges incident to any vertex of G is the same. When this constant sum is 0, we say that G is a zero-sum k-magic graph. The set of all k for which G is a zero-sum k-magic graph is the null set of G. In this paper, we will completely determine the null set of the join of a finite number of paths