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 
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/
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