Bipartite Handshake Lemma

Let $G=(L,R,E)$ where $L\cup R$ are its vertex set. Then,

$$\sum _{v\in L}\mathrm{deg}(v)=\sum _{v\in R}\mathrm{deg}(v)=|E|$$

Proof:

We know that each edge in a bipartite graph connects a vertex in $L$ to $R$. Thus, for each edge it has one endpoint in $L$. But then as ${\sum }_{v\in L}\mathrm{deg}(v)$ counts the edges with endpoints in $L$, then

$$\sum _{v\in L}\mathrm{deg}(v)=|E|$$

Likewise, as each edge has an endpoint in $R$,

$$\sum _{v\in R}\mathrm{deg}(v)=|E|$$

and so

$$|E|=\sum _{v\in L}\mathrm{deg}(v)=\sum _{v\in R}\mathrm{deg}(v)$$

Related: Handshake Lemma, Face Handshake Lemma.