On S-Magic Graphs

Let G=(V,E) be graph and let S be a set of positive integers with |S|=|V|. The graph G is said to be S-magic if there exists a bijection ϕ:V→S such that ∑u∈N(v)ϕ(u)=k, a constant, for all v∈V. We prove that if G is S-magic, then the corresponding magic constant is unique. We obtain several families of S-magic graphs. If M(G) denotes the set of all S-magic constants of G for different label sets S, we determine M(G) for several classes of graphs.