§ The handshaking lemma

§ Concrete situation:

Let's take a graph G(V,E)G \equiv (V, E). We can imagine that each edge has a potential of 22. We can redistribute this potential, by providing a potential of 11 to each of the vertices incident on the edge. This gives us the calculation that the total potential is 2E2|E|. But each vertex is assigned a potential of 11 for each edge incident on it. Thus, the total potential is vdegree(v)\sum_v \texttt{degree}(v). This gives the equality idegree(v)=2E\sum_i \texttt{degree}(v) = 2|E|. Thus, if each of the degrees are odd, considering modulo 2, the LHS becomes v1=V\sum_v 1 = |V| and the RHS becomes 00. Thus we have that V=0|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