Magic labellings of graphs over finite abelian groups

A total labelling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the set {1, 2, 3, . . . , v + e}. A labelling can be used to define a weight for each vertex and edge. For a vertex the weight is the sum of the label of the vertex and the labels of the incident edges. For an edge {x, y} the weight is the sum of the label of the edge and the labels of the end vertices x and y. A labelling is vertex-magic if all the vertices have the same weight. A labelling is edgemagic if all the edges have the same weight. A labelling is totally-magic if it is both vertex-magic and edge-magic. In this paper we generalize these concepts to A-labellings of a graph, that is labellings with the elements of an abelian group A of order v + e. We consider in detail A-labellings of star graphs.