§ The handshaking lemma
§ Concrete situation:
Let's take a graph . We can imagine that each edge has a potential
of . We can redistribute this potential, by providing a potential of
to each of the vertices incident on the edge. This gives us the calculation
that the total potential is . But each vertex is assigned a potential of
for each edge incident on it. Thus, the total potential is .
This gives the equality .
Thus, if each of the degrees are odd, considering modulo 2, the LHS becomes
and the RHS becomes . Thus we have that (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