# On γ-labelings of trees

• Volume: 25, Issue: 3, page 363-383
• ISSN: 2083-5892

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## Abstract

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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 $val\left(f\right)={\Sigma }_{e\in E\left(G\right)}{f}^{\text{'}}K\left(e\right)$. The maximum value of a γ-labeling of G is defined as $va{l}_{max}\left(G\right)=maxval\left(f\right):fisa\gamma -labelingofG$; while the minimum value of a γ-labeling of G is $va{l}_{min}\left(G\right)=minval\left(f\right):fisa\gamma -labelingofG$; The values $va{l}_{max}\left({S}_{p,q}\right)$ and $va{l}_{min}\left({S}_{p,q}\right)$ are determined for double stars ${S}_{p,q}$. We present characterizations of connected graphs G of order n for which $va{l}_{min}\left(G\right)=n$ or $va{l}_{min}\left(G\right)=n+1$.

## How to cite

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Gary Chartrand, et al. "On γ-labelings of trees." Discussiones Mathematicae Graph Theory 25.3 (2005): 363-383. <http://eudml.org/doc/270423>.

@article{GaryChartrand2005,
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 $val(f) = Σ_\{e ∈ E(G)\}f^\{\prime \}K(e)$. The maximum value of a γ-labeling of G is defined as $val_\{max\}(G) = max \{val(f) : f is a γ- labeling of G\}$; while the minimum value of a γ-labeling of G is $val_\{min\}(G) = min \{val(f) : f is a γ- labeling of G\}$; The values $val_\{max\}(S_\{p,q\})$ and $val_\{min\}(S_\{p,q\})$ are determined for double stars $S_\{p,q\}$. We present characterizations of connected graphs G of order n for which $val_\{min\}(G) = n$ or $val_\{min\}(G) = n+1$.},
author = {Gary Chartrand, David Erwin, Donald W. VanderJagt, Ping Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {γ-labeling; value of a γ-labeling; -labelling},
language = {eng},
number = {3},
pages = {363-383},
title = {On γ-labelings of trees},
url = {http://eudml.org/doc/270423},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Gary Chartrand
AU - David Erwin
AU - Donald W. VanderJagt
AU - Ping Zhang
TI - On γ-labelings of trees
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 3
SP - 363
EP - 383
AB - 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 $val(f) = Σ_{e ∈ E(G)}f^{\prime }K(e)$. The maximum value of a γ-labeling of G is defined as $val_{max}(G) = max {val(f) : f is a γ- labeling of G}$; while the minimum value of a γ-labeling of G is $val_{min}(G) = min {val(f) : f is a γ- labeling of G}$; The values $val_{max}(S_{p,q})$ and $val_{min}(S_{p,q})$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which $val_{min}(G) = n$ or $val_{min}(G) = n+1$.
LA - eng
KW - γ-labeling; value of a γ-labeling; -labelling
UR - http://eudml.org/doc/270423
ER -

## References

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1. [1] G. Chartrand, D. Erwin, D.W. VanderJagt and P. Zhang, γ-Labelings of graphs, Bull. Inst. Combin. Appl. 44 (2005) 51-68. Zbl1074.05079
2. [2] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. #DS6 (Oct. 2003 Version). Zbl0953.05067
3. [3] S.M. Hegde, On (k,d)-graceful graphs, J. Combin. Inform. System Sci. 25 (2000) 255-265. Zbl1219.05165

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