Geodesic
Arithmetic labelings and geometric labelings of countable graphs
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 539-544.

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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.
Keywords: arithmetic labeling of a graph, geometric labeling of a graph 
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Vijayakumar, Gurusamy. Arithmetic labelings and geometric labelings of countable graphs. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 539-544. http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a0/

[1] B.D. Acharya and S.M. Hegde, Arithmetic graphs, J. Graph Theory 14 (1990) 275-299, doi: 10.1002/jgt.3190140302.

[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.

[3] L.W. Beineke and S.M. Hegde, Strongly multiplicative graphs, Discuss. Math. Graph Theory 21 (2001) 63-75, doi: 10.7151/dmgt.1133.

[4] S.M. Hegde, On multiplicative labelings of a graph, J. Combin. Math. and Combin. Comp. 65 (2008) 181-195.

[5] S.M. Hegde and P. Shankaran, Geometric labeled graphs, AKCE International J. Graphs and Combin. 5 (2008) 83-97.

[6] G.R. Vijayakumar, Arithmetic labelings and geometric labelings of finite graphs, J. Combin. Math. and Combin. Comp. (to be published).

[7] D.B. West, Introduction to Graph Theory, Second edition (Printice Hall, New Jersey, USA, 2001).