Discussiones Mathematicae Graph Theory 30(1) (2010) 5-15
doi: 10.7151/dmgt.1472

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VALUE SETS OF GRAPHS EDGE-WEIGHTED WITH ELEMENTS OF A FINITE ABELIAN GROUP

Edgar G. DuCasse1, Michael L. Gargano2,*, Louis V. Quintas1

Pace University
One Pace Plaza, New York, NY 10038, USA
1Mathematics Department
2Mathematics and Computer Science Departments
e-mail: educasse@pace.edu, mgargano@pace.edu, lquintas@pace.edu
*Michael L. Gargano passed away on May 31, 2008

Abstract

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.

Keywords: graph labeling, edge labeling, vertex labeling, abelian group.

2010 Mathematics Subject Classification: 05C78, 05C22, 05C25.

References

[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).
[2] J.A. Gallian, A dynamic survey of graph labeling, Electronic J. Combin. 14 (2007) #DS6.
[3] J.A. Gallian, Contemporary Abstract Algebra, 6th Edition (Houghton-Mifflin, Boston, Massachusetts, 2006).
[4] D.B. West, Introduction to Graph Theory, 2nd Edition (Prentice Hall, Upper Saddle River, New Jersey, 2001).

Received 23 May 2008
Revised 5 January 2009
Accepted 5 January 2009