Degree sum formula

Question

Suppose the G = (V,E) is a connected graph with n vertices and n-1 edges. Use the degree-sum formula for vertices to prove that G has a vertex of degree 1.

I can't seem to find where to start in this.

Any leads would be appreciated

Answer 1

We note that the statement is false for a graph with one vertex and no edges,so we shall prove the statement for $n>1.$ By the degree-sum formula, $$\sum_{v\in V}{\deg(v)}=2(n-1).$$ Since the sum has $n$ term, there must be a $v_0\in V$ such that $\deg(v_0) < 2$ as otherwise, we would have $\sum{\deg(v)\ge 2n > 2(n-1).}$ We cannot have $\deg(v_0)=0,$ since $G$ is a connected graph with more than one vertex, so $\deg(v_0)=1.$

Note that under the hypotheses, $G$ actually has at least two vertices of degree $1.$ Can you continue the proof to show that?