Geodesic
Total edge irregularity strength of trees
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 449-456

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A total edge-irregular k-labelling ξ:V(G)∪ E(G) → 1,2,...,k of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree Δ on p vertices
Keywords: graph labelling, tree, irregularity strength, total labellings, total edge irregularity strength 
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Ivančo, Jaroslav; Jendrol', Stanislav. Total edge irregularity strength of trees. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 449-456. http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a9/