## § The handshaking lemma

#### § Concrete situation:

Let's take a graph $G \equiv (V, E)$. We can imagine that each edge has a potential
of $2$. We can redistribute this potential, by providing a potential of $1$
to each of the vertices incident on the edge. This gives us the calculation
that the total potential is $2|E|$. But each vertex is assigned a potential of $1$
for each edge incident on it. Thus, the total potential is $\sum_v \texttt{degree}(v)$.
This gives the equality $\sum_i \texttt{degree}(v) = 2|E|$.
Thus, if each of the degrees are odd, considering modulo 2, the LHS becomes
$\sum_v 1 = |V|$ and the RHS becomes $0$. Thus we have that $|V| = 0$ (mod 2), or
the number of vertices remains even.
I learnt of this nice way of thinking about it in terms of potentials when
reading a generalization to simplicial
complexes.
#### § References