# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## ON γ-LABELINGS OF TREES

 Gary Chartrand Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA David Erwin School of Mathematical Sciences University of KwaZulu-Natal Durban 4041, South Africa Donald W. VanderJagt Department of Mathematics Grand Valley State University Allendale, MI 49401 USA Ping Zhang Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA

## Abstract

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G)→{0,1,2,...,m} that induces a labeling f ':E(G)→{1,2,...,m} of the edges of G defined by f '(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is val(f) = Σe ∈ E(G)f '(e). The maximum value of a γ-labeling of G is defined as
valmax(G) = max { val(f) : f is a γ− labeling of G};
while the minimum value of a γ-labeling of G is
valmin(G) = min { val(f) : f is a γ− labeling of G};

The values valmax(Sp,q) and valmin(Sp,q) are determined for double stars Sp,q. We present characterizations of connected graphs G of order n for which valmin(G) = n or valmin(G) = n+1.

Keywords: γ-labeling, value of a γ-labeling.

2000 Mathematics Subject Classification: 05C78, 05C05.

## References

 [1] G. Chartrand, D. Erwin, D.W. VanderJagt and P. Zhang, γ-Labelings of graphs, Bull. Inst. Combin. Appl. 44 (2005) 51-68. [2] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. #DS6 (Oct. 2003 Version). [3] S.M. Hegde, On (k,d)-graceful graphs, J. Combin. Inform. System Sci. 25 (2000) 255-265.