A proper edge coloring of a graph is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor-distinguishing index of is the minimum number of colors in a neighbor-distinguishing edge coloring of .
Let be a graph with maximum degree and without isolated edges. In this paper, we prove that if , and if . This improves a result in Zhang et al. (2014), which states that for any graph without isolated edges. Moreover, we prove that if is a semi-regular graph (i.e., each edge of is incident to at least one -vertex), then .