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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1472,
author = {Edgar G. DuCasse and Michael L. Gargano and Louis V. Quintas},
title = {Value sets of graphs edge-weighted with elements of a finite abelian group},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {5-15},
zbl = {1215.05149},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1472}
}
Edgar G. DuCasse; Michael L. Gargano; Louis V. Quintas. Value sets of graphs edge-weighted with elements of a finite abelian group. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 5-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1472/
[000] [1] E.G. DuCasse, M.L. Gargano, and L.V. Quintas, The edge-weight sums of a graph mod n (to have been presented at Thirty-Ninth Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, Boca Raton, Florida, March 3-7, 2008 by Michael L. Gargano, who could not do so due to illness).
[001] [2] J.A. Gallian, A dynamic survey of graph labeling, Electronic J. Combin. 14 (2007) #DS6. | Zbl 0953.05067
[002] [3] J.A. Gallian, Contemporary Abstract Algebra, 6th Edition (Houghton-Mifflin, Boston, Massachusetts, 2006). | Zbl 0972.00001
[003] [4] D.B. West, Introduction to Graph Theory, 2nd Edition (Prentice Hall, Upper Saddle River, New Jersey, 2001).