Generalizations of magic graphs

A graph is magic if the edges are labeled with distinct nonnegative real numbers such that the sum of the labels incident to each vertex is the same. Given a graph finite G, an Abelian group $\text{g}$, and an element r(v) ∈ $\text{g}$ for every v ∈ V(G), necessary and sufficient conditions are given for the existence of edge labels from $\text{g}$ such that the sum of the labels incident to v is r(v). When there do exist labels, all possible labels are determined. The matroid structure of the labels is investigated when $\text{g}$ is an integral domain, and a dimensional structure results. Characterizations of several classes of graphs are given, namely, zero magic, semi-magic, and trivial magic graphs.