Title: An upper bound on the distinguishing index of graphs with minimum degree at least two,
Abstract: 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. It was conjectured by Pilśniak (2015) that for any
2-connected graph $D'(G) \leq \lceil \sqrt{\Delta (G)}\rceil +1$. We prove a
more general result for the distinguishing index of graphs with minimum degree
at least two from which the conjecture follows. Also we present graphs $G$ for
which $D'(G)\leq \lceil \sqrt{\Delta }\rceil$.