  A {\em strong edge coloring} of a graph $G$ is a proper edge coloring in
which every color class is an induced matching. The {\em strong chromatic
index} $\chiup_{s}'(G)$ of a graph $G$ is the minimum number of colors in a
strong edge coloring of $G$. In this note, we improve a result by D{\k e}bski
\etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show
that the strong chromatic index of a $k$-degenerate graph $G$ is at most
$(4k-2) \cdot \Delta(G) - 2k^{2} + 1$. As a direct consequence, the strong
chromatic index of a $2$-degenerate graph $G$ is at most $6\Delta(G) - 7$,
which improves the upper bound $10\Delta(G) - 10$ by Chang and Narayanan
[Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2)
119--126]. For a special subclass of $2$-degenerate graphs, we obtain a better
upper bound, namely if $G$ is a graph such that all of its $3^{+}$-vertices
induce a forest, then $\chiup_{s}'(G) \leq 4 \Delta(G) -3$; as a corollary,
every minimally $2$-connected graph $G$ has strong chromatic index at most $4
\Delta(G) - 3$. Moreover, all the results in this note are best possible in
some sense.
