Abstract
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 g , and an element r(v) ∈ g for every v ∈ V( G), necessary and sufficient conditions are given for the existence of edge labels from 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 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.
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