Graphs with degree $\Delta$ have a maximum independent set of size $\alpha \geqslant \frac{n}{\Delta}$ where $n$ is the number of vertices. But, are there graphs such that $\alpha \approx \frac{n}{\Delta}$, or even $\alpha = c \frac{n}{\Delta}$ for some constant $c \leqslant 1$?
Graphs with bounded degree and small max independent set
graphs