Proposition 10.6 -- the commute time identity

Let $c$ be a network on a finite vertex set $V$: a symmetric nonnegative conductance function with total vertex conductance $c(x)=\sum_yc(x,y)>0$ everywhere, carrying the irreducible walk $P(x,y)=c(x,y)/c(x)$. Write $c_G=\sum_xc(x)$ for the total conductance of the network, $\mathbb E_a(\tau_b)$ for the expected number of steps for the walk started at $a$ to first reach $b$, and $R(a\leftrightarrow b)$ for the effective resistance, defined through the voltage $W(x)=\mathbb P_x\{\tau_a<\tau_b\}$ and current $\|I\|=\sum_yc(a,y)[W(a)-W(y)]$ as $R(a\leftrightarrow b)=\|I\|^{-1}$.

The theorem (the Commute Time Identity, Proposition 10.6 of Levin–Peres–Wilmer, the capstone of Chapters 9–11) asserts: for any two distinct vertices $a\ne b$,

The expected round-trip time between two vertices is exactly the total conductance times the effective resistance between them. This single identity converts the entire electrical toolkit — series/parallel reduction, Thomson's principle, Rayleigh monotonicity — into exact computations and bounds for hitting and cover times of reversible chains.