I know that a connected graph on n vertices has at least n-1 edges so n-1 edges would be a sufficient condition, but is it necessary? I can't seem to think of a counterexample but don't want to assume that it must therefore be true without a solid proof.
Any pointers towards the right direction would be much appreciated. Thanks in advance!
The graph with a cycle of length $n-1$ and an isolated vertex has $n$ vertices and $n-1$ edges but is not connected.