Trying to solve this proof: proof any undirected loop-free graph on n vertices with at least (n-1)(n-2)/2 + 1 edges is connected. Give an example of a disconnected n-vertex graph with one fewer edge
I am not sure how to solve this one, I know that graphs with (n-1)(n-2)/2 edges are connected and then have assumed adding one edge would also leave this graph connected. When i prove by contradiction i end up with a disconnected graph unable to have k + 1 vertices and at least k(k-1)/2 + 1 edges, but feel that this is incorrect.
How to prove this and how to give an example with one fewer edge?