Suppose you have $n$ points that you will connect with $m$ edges. There must be a path from each point to every other point (not directly, but it must be possible to get there somehow), and there should not be more than one edge for any pair of points.
How many connections can be formed with $n$ points and $m$ edges given these constraints?
$m$ will always be at least $n-1$ and no greater than $\frac{n\cdot (n-1)}{2}$.
Example:
Suppose $n$ is $3$ (points $a$, $b$, and $c$, and $m$ is $2$). Then there would be $3$ possible graphs:
$a$ -- $b$ -- $c$
$a$ -- $c$ -- $b$
$c$ -- $a$ -- $b$
I've yet to find a mathematical formula that will let me generalize this for $n$ and $m$, but intuitively, after solving for all cases of $n = 3$ and $n = 4$, it seems like there should be one. Is it possible?
