 Commute time of a random walk

For a finite connected undirected graph, the commute time of a random walk between $a$ and $b$ is the expected time to travel from $a$ to $b$ and back:For the simple random walk and unit edge resistances, it satisfies $C_{ab}=2|E|R_{\mathrm{eff}}(a,b)$, where $R_{\mathrm{eff}}$ is effective resistance. Each additional parallel route lowers resistance and therefore lowers commute time when the total edge count is fixed.^[ref]

On a path with $m$ edges, the endpoint resistance is $m$ and the commute time is $2m^2$; each one-way hitting-time expectation is $m^2$. On a triangle, two routes give resistance $1\parallel 2=2/3$, so the commute time between any pair is $2\cdot 3\cdot (2/3)=4$. The identity applies to this reversible walk and its associated resistor network, not to arbitrary directed transition rules.