Handshaking lemma

Statement

Suppose $G$ is a finite undirected graph. The handshaking lemma says that the quantities (1)-(4) are all equal. Note that the equality of all the quantities apart from (1) is obvious, and the main content of the lemma is the equality with (1).

1. Twice the number of edges of $G$.
2. The sum of the degrees of all vertices of $G$.
3. The sum of the elements of the degree sequence of $G$.
4. The trace of the degree matrix of $G$.

In particular, because the quantity defined by (1) is clearly an even nonnegative integer, it also shows that all the other quantities are even nonnegative integers.