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Ashok K. Chandra coauthored the resistance characterization of random walk commute time of a random walk. The identity $H_{st}+H_{ts}=2m{R}_{st}$ relates expected travel times on a graph to effective resistance in its unit-resistor network.^[ref]

For a finite connected loopless undirected electrical network without killing, give each edge a positive electrical conductance $c_e$, put $W={\sum }_e{c}_e$, and use transition probabilities $P(x,y)=c_{xy}/c_x$, where $c_x={\sum }_y{c}_{xy}$. Then the commute time identity isThe factor is total electrical conductance, not total electrical resistance. A killed random walk can fail to reach its target and is outside this statement.

Here is a direct proof. Let $h_z(x)={\mathbb{E}}_x{T}_z$ and define the network operator $Lh(x)={\sum }_y{c}_{xy}(h(x)-h(y))$. First-step analysis gives $L{h}_z(x)=c_x$ off $z$. Summing $L{h}_z$ over all graph vertices gives zero, so $L{h}_z(z)=c_z-2W$. Therefore $u=(h_z-h_a)/(2W)$ has divergence $+1$ at $a$, $-1$ at $z$, and zero elsewhere. Its voltage drop is $(h_z(a)+h_a(z))/(2W)$, which is the effective resistance for unit electric current.

For an unweighted path with $n$ edges, $W=n$ and endpoint effective resistance is $n$, giving commute time $2n^2$. Uniformly multiplying all electrical conductances changes $W$ and effective resistance inversely and leaves the Markov chain unchanged. ^[ref]

Prabhakar Raghavan coauthored an electrical-network method for studying random walks. Their exact commute time of a random walk identity and resistance bounds on cover time connect the geometry of a graph with the time required to explore it.^[ref]

Prasoon Tiwari coauthored the connection between effective resistance and random walk times. For an undirected graph with $m$ unit-resistance edges, the commute time of a random walk between two vertices equals $2m$ times their effective resistance.^[ref]

Roman Smolensky contributed to the electrical-network analysis of random walks on graphs. The joint work identifies commute time of a random walk with $2mR$, translating a stochastic round trip into an effective resistance calculation.^[ref]

Walter L. Ruzzo coauthored bounds linking random walk cover time to electrical-network resistance. The paper also proves the exact commute time of a random walk identity, enabling deterministic effective resistance calculations to control expected exploration times.^[ref]