[For the entire collection see Zbl 0698.00017.]
A weighting \(w\) of a simple graph \(G\) is a function that assigns to each edge of \(G\) a positive integer. The weighting \(w\) is said to be admissible if the sums of the weights at each vertex are all distinct. This paper is concerned with the evaluation of the parameter \(i(G)\), which is the minimum cardinality of the image of \(w\) over all admissible weightings \(w\) of \(G\). The authors evaluate \(i(G)\) for complete graphs, complete bipartite graphs, cycles, paths, irreducible trees, and several other families of graphs. They also prove that if \(G\) is a graph of order \(n\) such that \(i(G)\) is finite, then \(i(G)\leq n-1\), except that \(i(K_ 3)=3\).