Theorem 3.1

If $G$ is a graph with vertices $v_1,v_2,\dots ,v_n$, then $\sum _{i=1}^{n}d(v_i)=2|E(G)|$

Proof

Each loop contributes 2 to the summation. In the summation, each edge $e=uv$ that is not a loop is counted exactly twice, once in $d(u)$ and once in $d(v)$