How many induced subgraphs does $G$ have of specific size Let $G$ be a connected graph with $n$ vertices and $O(nk)$ edges, ($k$ is parameter). 
Then how many connected induced subgraphs of size $l$  does $G$ have?   

The simple case when $l=2$, The number of subgraphs is number of edges which is $O(nk)$. How can we generalize this one.
Consider a star graph, if we take a center of star union with $l-1$ arbitrary vertices then the corresponding subgraph is induced connected subgraph. So the maximum number is about $n-1\choose l-1$. We can extend this class of graphs by connecting arbitrary vertices together to have more edges. It's also possible to achieve closer to $n \choose l$ if $l$ is much smaller than $k$, or $k$ is big enough.