effective resistance

The effective resistance of an electrical network $G = (V, E)$ .

R_{e f f} = \frac{U_{1} - U_{n}}{I_{b a t}}

where $U_{i}$ is the potential function at $i$ .

If we make the assumption that $U_{n} = 0$ , and $I_{b a t} = 1$ (grounding and normalizing the battery), then it just suffices to find $U_{1}$ . This can be done via Cramer's rule:

U_{1} = \frac{det \tilde{\tilde{K}}}{det \tilde{K}}

where $\tilde{\tilde{K}}$ is obtained by removing the 1st and $n$ th rows and columns of $K$ , and $\tilde{K}$ is obtained by removing the $n$ th row and column of $K$ .

Equivalently, thinking about what the graph of $\tilde{\tilde{K}}$ is, we notice that this is the Laplacian of $G^{'}$ obtained from $G$ by gluing together the 1st and $n$ th vertex. Then, applying weighted matrix tree theorem, we have

R_{e f f} = \frac{\sum_{T spans G^{'}} wt (T)}{\sum_{T spans G} wt (T)}