Q:

Number of ways n-1 edges may be chosen to form a connected graph with n nodes

Basically, you require n-1 edges, to make a connected graph with n nodes.  I would like to know if there is any theory behind finding the number of distinct ways you can select the n-1 edges, from the total n(n-1)/2 edges that are possible, such that the graph remains connected. 

A:

There are exactly nn-2 connected graphs with vertex set {1,...n} for n > 0.   This result is known as Cayley's Formula.