Effective Resistance and Commute Time on the 4-Cycle

Consider the cycle graph $C_4$ with vertices $\{0,1,2,3\}$ arranged in a square, and each edge having unit resistance.

(a) Compute the effective resistance $R_{\mathrm{eff}}(0, 2)$ between the two diagonally opposite vertices.

(b) Using the commute-time identity $C(u,v) = 2m \cdot R_{\mathrm{eff}}(u,v)$, find the commute time $C(0,2)$.

(c) By exploiting the symmetry $0 \leftrightarrow 2$ of $C_4$, determine $h(0 \to 2)$ and verify consistency with the commute time.