an upper bound on the distinguishing index of graphs with minimum degree at least two

The distinguishing index of a simple graph g, denoted by d′(g), is the least number of labels in an edge labeling of g not preserved by any non-trivial automorphism. we prove that for a connected graph g with maximum degree δ, if the minimum degree is at least two, then d′(g)≤г√δ˥+1. we also present graphs g for which d′(g)≤г √δ(g)˥.