Effective Resistance and Commute Time on the Hypercube Q₃

Consider the $3$-dimensional hypercube graph $Q_3$ (vertices are binary strings of length $3$; edges connect strings differing in exactly one bit). Each edge has unit resistance.

(a) Using the symmetry of $Q_3$, compute the effective resistance $R_{\mathrm{eff}}(000, 111)$ between the two antipodal vertices.

(b) The commute time $C(u, v)$ of a random walk between vertices $u$ and $v$ on a graph satisfies $C(u, v) = 2m \cdot R_{\mathrm{eff}}(u, v)$, where $m$ is the number of edges. Use this to find the commute time between $000$ and $111$.