Value sets of graphs edge-weighted with elements of a finite abelian group

Given a graph G = (V, E) of order n and a finite abelian group H = (H, +) of order n, a bijection f of V onto H is called a vertex H-labeling of G. Let g(e) ≡ (f(u) + f(v)) mod H for each edge e = {u, v} in E induce an edge H-labeling of G. Then, the sum Hvalf (G) ≡ ∑ e∈E g(e) mod H is called the H-value of G relative to f and the set HvalS(G) of all H-values of G over all possible vertex H-labelings is called the H-value set of G. Theorems determining HvalS(G) for given H and G are obtained.