Rayleigh monotonicity law

Remarks

By Thomson's principle, each effective resistance is the minimum energy of a unit flow from $a$ to $b$. For every unit flow $\theta$, the energy $\sum_e r(e) \theta(e)^2$ is at most $\sum_e r'(e) \theta(e)^2$. So the minimum for $r$ is at most the minimum for $r'$. Example: the path $a - m - b$ with unit resistances has $R_{\mathrm{eff}}(a, b) = 2$. Adding an edge from $a$ to $b$ of resistance $1$ gives $R_{\mathrm{eff}}(a, b) = \tfrac23$. This agrees with the law, viewing the new edge as the limit of an edge of very large resistance whose resistance is lowered. Two consequences are standard: removing an edge, as long as the graph stays connected, does not decrease effective resistances. Gluing two vertices into one, which amounts to giving a resistance $0$ to an edge between them, does not increase effective resistances.

Going deeper

For a student

The law is the main tool for comparing random walks on different graphs. To show that a walk on a graph is recurrent, one may glue vertices together, which lowers resistances, and show that the resistance to infinity is still infinite. To show transience, one may delete edges, which raises resistances, and find a subgraph, such as a tree, with finite resistance to infinity. Combined with the criterion that relates recurrence to the resistance to infinity, this proves the transience of $\Z^3$ from the transience of a suitable subgraph.