Relationship between the distinguishing index, minimum degree and maximum degree of graphs

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a 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. Motivated by a conjecture by Pil\'sniak (2017) that implies that for any $2$-connected graph $D'(G) \leq \lceil \sqrt{\Delta (G)}\rceil +1$, we prove that for any graph $G$ with $\delta\geq 2$, $D'(G) \leq \lceil \sqrt[\delta]{\Delta }\rceil +1$. Also, we show that the distinguishing index of $k$-regular graphs is at most $2$, for any $k\geq 5$.