Graphs with bounded degree and small max independent set
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.