Graphs with bounded degree and small max independent set 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 $c \leqslant 1$?
First of all, the correct bound is $n/(\Delta + 1)$. This is tight for unions of cliques of size $\Delta+1$. If you want a connected graph, take a union of $\Delta$-cliques and connect them in a path or a cycle. This gives $n/\Delta$, so there is a small loss over the lower bound $n/(\Delta+1)$. If you want a connected regular graph, add a matching to the previous graph.