[Math] Is it possible in a group of seven people for each to be friends with exactly 3 others

Question

Is it possible in a group of seven people for each to be friends with exactly 3 others?

I know that the sum of degrees of vertices in a graph must be even.

Answer 1

Let $G(V,E)$ represent our graph. Because we know every person must be friends with excatly three other people, we now: $\forall v \in V$. deg$(v) = 3$. By the hand-shaking lemma: \begin{align} 2|E| = \sum_{v \in V} deg(v) \Rightarrow 2|E| = \sum_{n=1}^7 3 = 21. \end{align}

However $|E|$ is an integer so we come to a contradiction, since we get $|E| = \frac{21}{2}.$