If $G$ is an acyclic graph, How can we prove that $G$ is connected?

Question

How to Prove $G$ is connected, if $G$ is an acyclic graph on $n \ge 1$ vertices containing exactly $n − 1$ edges?

Answer 1

Induction on $n$ is a good option. Hint:

If $|G| \ge 2$ and $G$ is acyclic and $G$ has at least one edge then $G$ has a vertex of degree 1.

Answer 2

If $G$ is acyclic with $n$ vertices and $n-1$ edges then $G$ is a connected tree.

Proof: (by induction on $n$)

If $n=1$ (or $n=2$), the result is trivial.

Suppose the property holds for a given $n$, and consider an acyclic graph with $n+1$ vertices and $n$ edges. Since $G$ is acyclic, there is at least one vertex $v$ with degree $1$. The graph induced by deleting this node is acyclic with $n$ vertices and $n-1$ edges: it is a connected tree by assumption. Linking $v$ back to the tree does not create a cycle, it follows that $G$ is a connected tree.