Kirchhoffs laws as described by homology

Question

I was wondering what the relation between Kirchhoffs laws and simplicial homology is. The voltage law states that $\sum V = 0$ around a loop, and the current law that $\sum I = 0$ around a vertex, so it seems the voltage law is described as a statement about 1-chains, and current about 0-cochains?

The question is: How exactly would one describe these laws in terms of the boundary and coboundary operators, $\partial$ and $\delta$?

Answer 1

Let $G$ be your graph.

I think the statements you're looking for are:

• $V$ is a $1$-coboundary (with real values); that is, it is a $1$-cochain that can be expressed as $V = \delta U$ for some $0$-cochain $U$. (this is not a literal translation, of course, but a consequence; I think something like $V$ representing the zero class in $H^1(G; \mathbb{R})$ is closer)
• $I$ is a $1$-cycle; that is, a $1$-chain (with real coefficients) such that $\partial I = 0$.