This paper is dedicated to studying the following question: Is it always possible to injec-tively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K 2), so that every two adjacent vertices of G get distinguished by their sums of incident weights? We answer positively to this question for some classes of graphs, such as trees and regular graphs, while, for some other classes of graphs, such as 2-degenerate graphs and graphs with maximum average degree at most 3, we prove that, provided we use a constant number of additional edge weights, the claimed assignment always exists. Our investigations here, are strongly related to several aspects of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture.