Counting the number of directed graphs with $N$ vertices and $E$ edges? Does any body who has good back ground in graph theory tells me that how many possible directed graphs will be there with $N$ vertices and $E$ edges. I need all the possible combinations even even isomorphic graphs also.

If you treat isomorphic graphs separately, then one way to think about this is as follows. If you start off with $N$ nodes and want $E$ edges, you could build a graph by considering an $E$-element subset of the set of all possible edges in the graph. The number of possible edges is $N^2$, since each node can be connected to each other node (including itself), so the number of ways you can pick $E$ of these edges is $\binom{N^2}{E}$. This is

$$ \frac{(N^2)!}{(E!) \cdot (N^2 - E)!}, $$

which is a staggeringly huge number.

Hope this helps!