Short version:
If you take n vertices and connect them together with lines, you'll have nā1 lines, why nā1? Other than the obvious visual proof, is there something that could be more, I don't know, substantial?
O - dot
O------------------------O------------------------O Three dots, two lines. $n$ -> $n-1$
Longer version:
Other than the obvious there's three dots, there are two lines - therefore $n-1$ is there some sort of a proof? For example, I can easily see how $n$ disconnected dots give $n/2$ lines, because there must be $m$ groups of 2 dots or $m$ lines.
I've been thinking about this a bit, if there's n dots, only the first line requires two unique dots and every subsequent one requires only 1 additional. But I still don't see the $n-1$ in that. Please assist me with this triviality, thanks!
Dots/vertices, however you want to name them.