Geodesic
Value sets of graphs edge-weighted with elements of a finite abelian group
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 5-15

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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 Hval_f(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.
Keywords: graph labeling, edge labeling, vertex labeling, abelian group 
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DuCasse, Edgar; Gargano, Michael; Quintas, Louis. Value sets of graphs edge-weighted with elements of a finite abelian group. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 5-15. http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a0/