Arithmetic labelings and geometric labelings of countable graphs

Gurusamy Rengasamy Vijayakumar

Discussiones Mathematicae Graph Theory (2010)

  • Volume: 30, Issue: 4, page 539-544
  • ISSN: 2083-5892

Abstract

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An injective map from the vertex set of a graph G-its order may not be finite-to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is called arithmetic (geometric) if it admits an arithmetic (a geometric) labeling. In this article, we show that the two notions just mentioned are equivalent-i.e., a graph is arithmetic if and only if it is geometric.

How to cite

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Gurusamy Rengasamy Vijayakumar. "Arithmetic labelings and geometric labelings of countable graphs." Discussiones Mathematicae Graph Theory 30.4 (2010): 539-544. <http://eudml.org/doc/270872>.

@article{GurusamyRengasamyVijayakumar2010,
abstract = {An injective map from the vertex set of a graph G-its order may not be finite-to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is called arithmetic (geometric) if it admits an arithmetic (a geometric) labeling. In this article, we show that the two notions just mentioned are equivalent-i.e., a graph is arithmetic if and only if it is geometric.},
author = {Gurusamy Rengasamy Vijayakumar},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {arithmetic labeling of a graph; geometric labeling of a graph},
language = {eng},
number = {4},
pages = {539-544},
title = {Arithmetic labelings and geometric labelings of countable graphs},
url = {http://eudml.org/doc/270872},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Gurusamy Rengasamy Vijayakumar
TI - Arithmetic labelings and geometric labelings of countable graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 4
SP - 539
EP - 544
AB - An injective map from the vertex set of a graph G-its order may not be finite-to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is called arithmetic (geometric) if it admits an arithmetic (a geometric) labeling. In this article, we show that the two notions just mentioned are equivalent-i.e., a graph is arithmetic if and only if it is geometric.
LA - eng
KW - arithmetic labeling of a graph; geometric labeling of a graph
UR - http://eudml.org/doc/270872
ER -

References

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  1. [1] B.D. Acharya and S.M. Hegde, Arithmetic graphs, J. Graph Theory 14 (1990) 275-299, doi: 10.1002/jgt.3190140302. Zbl0735.05071
  2. [2] B.D. Acharya and S.M. Hegde, On certain vertex valuations of a graph, Indian J. Pure and Appl. Math. 22 (1991) 553-560. Zbl0759.05086
  3. [3] L.W. Beineke and S.M. Hegde, Strongly multiplicative graphs, Discuss. Math. Graph Theory 21 (2001) 63-75, doi: 10.7151/dmgt.1133. Zbl0989.05101
  4. [4] S.M. Hegde, On multiplicative labelings of a graph, J. Combin. Math. and Combin. Comp. 65 (2008) 181-195. Zbl1180.05103
  5. [5] S.M. Hegde and P. Shankaran, Geometric labeled graphs, AKCE International J. Graphs and Combin. 5 (2008) 83-97. Zbl1171.05408
  6. [6] G.R. Vijayakumar, Arithmetic labelings and geometric labelings of finite graphs, J. Combin. Math. and Combin. Comp. (to be published). Zbl1232.05213
  7. [7] D.B. West, Introduction to Graph Theory, Second edition (Printice Hall, New Jersey, USA, 2001). 

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