Geodesic
The Distance Magic Index of a Graph
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 135-142.

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Let G be a graph of order n and let S be a set of positive integers with |S| = n. Then G is said to be S-magic if there exists a bijection ϕ : V (G) → S satisfying Σ_ x ∈ N (u) ϕ (x) = k (a constant) for every u ∈ V (G). Let α (S) = max{ s : s ∈ S }. Let i(G) = min α (S), where the minimum is taken over all sets S for which the graph G admits an S-magic labeling. Then i(G) − n is called the distance magic index of the graph G. In this paper we determine the distance magic index of trees and complete bipartite graphs.
Keywords: distance magic labeling, distance magic index, S -magic graph, S -magic labeling 
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Godinho, Aloysius; Singh, Tarkeshwar; Arumugam, S. The Distance Magic Index of a Graph. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 135-142. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a10/

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