Effective resistance

Remarks

The current $I$ is positive: $0 \le v \le 1$ by the maximum principle, and $v(y) < 1$ for some neighbour $y$ of $a$ since $G$ is connected. Scaling $v$ shows that $R_{\mathrm{eff}}(a, b)$ is the voltage difference between $a$ and $b$ when a current of strength $1$ enters at $a$ and leaves at $b$. Example: for the path $a - m - b$ with unit conductances, $v(m) = \tfrac12$, so $I = \tfrac12$ and $R_{\mathrm{eff}}(a, b) = 2$, the sum of the two resistances in series. If an edge of conductance $1$ is added between $a$ and $b$, the current becomes $\tfrac12 + 1$, so $R_{\mathrm{eff}}(a, b) = \tfrac23$, as for resistors in parallel. Non-example: if $a$ and $b$ lie in different components of a disconnected graph, no current can flow between them. Then the effective resistance is not a finite number, which is why connectedness is part of the definition of a network. With $L$ the weighted graph Laplacian and $L^{+}$ its Moore–Penrose pseudoinverse, $R_{\mathrm{eff}}(a, b) = (e_a - e_b)^{T} L^{+} (e_a - e_b)$, where $e_x$ is the indicator vector of $x$. The function $(a, b) \mapsto R_{\mathrm{eff}}(a, b)$, extended by $0$ on the diagonal, is a metric on $V$, the resistance distance.