Induced Matchings in Graphs of Bounded Maximum Degree

For a graph $G$, let $\nu_s(G)$ be the induced matching number of $G$. We prove that $\nu_s(G) \geq \frac{n(G)}{(\lceil\frac{\Delta}{2}\rceil+1) (\lfloor\frac{\Delta}{2}\rfloor+1)}$ for every graph of sufficiently large maximum degree $\Delta$ and without isolated vertices. This bound is sharp. Moreover, there is polynomial-time algorithm which computes induced matchings of size as stated above.