# Suppose G is simple graph with n vertices. Prove that G has twice as many edges as vertices only if $n\geq 5$

I just didn't understand the second part of the prove

"G has twice as many edges as vertices only "... what do I actually have to prove ?

I understand it like $n=2e$ , is it right ?

then isn't it's just similar to the degree equation ??

• Note the 'only if'. All that you have to show is that a simple graph of strictly less than $5$ vertices cannot have $2n$ vertices. – Discrete lizard May 6 '18 at 16:47
• Will do , thanks for your time, and you mean 2n edges right ? @Discretelizard – NANA May 6 '18 at 16:53
• Indeed, I mean $2n$ edges, of course. – Discrete lizard May 6 '18 at 17:25

When $n \leq 4$, the possible number of edges is less than twice the number of vertices. For example, when $n = 4$, there are 4 vertices but only 6 edges. In contrast, the complete graph on 5 vertices contains 10 edges.