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)≤⌈Δ−−√⌉+1. We also present graphs G for which D′(G)≤⌈Δ(G)−−−−−√⌉.