The integer-magic spectra and null sets of the cartesian product of trees

Let A be a non-trivial, finitely-generated abelian group and A* = A\{0}. A graph is A-magic if there exists an edge labeling f using elements of A* which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called the A-magic value. The integer-magic spectrum of a graph G is the set (Formula presented), where N is the set of natural numbers. The null set of G is the set of integers k ∈ N such that G has a Z_k-magic labeling with magic value 0. In this paper, we determine the integer-magic spectra and null sets of the Cartesian product of two trees.