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. [ABSTRACT FROM AUTHOR]