graph theory
A simple undirected graph has no self-loops and no parallel edges.
Determine the number of simple undirected graphs $G = (V, E)$ with $V = {1, . . . , n}$
Also, how can I find the number of simple graphs with vertices of degree 1?
Does someone knows a traditional method to solve this? Please help.
$\phantom{30 characters for acceptance.......}$
This is a combination problem, as the order of the edges doesn't matter.
For an undirected graph with $n$ vertices, there are $\frac{n!}{2!(n-2)!}$ or $\frac{n(n-1)}{2}$ possible edges. You could also say, "$n$ choose $2$" possible edges. Of these possible edges, we are choosing $m$ of these edges. Therefore, there are computing "$n$ choose $2$ choose $m$".
http://www.wolframalpha.com/input/?i=(n+choose+2)+choose+m
In your case, (7 choose 2) choose 10 = 352716
Best Answer
Assuming you got $N$ vertices and $M$ edges, then since you got $N$ total vertices, which means that you got $$\sum_{k=1}^{N-1} k = \frac{N(N-1)}{2} = P$$ possible edges. Now out of those $P$, pick the $M$ that are present, i.e. $P \choose M$ :).