Effective Resistance and Commute Time on K₂,₃

Consider the complete bipartite graph $K_{2,3}$ with parts $A = \{a_1, a_2\}$ (each of degree $3$) and $B = \{b_1, b_2, b_3\}$ (each of degree $2$). Each edge has unit resistance.

(a) Compute the effective resistance $R_{\mathrm{eff}}(a_1, a_2)$ between the two vertices of part $A$.

(b) Use the commute-time identity $C(u,v) = 2m \cdot R_{\mathrm{eff}}(u,v)$ to find the expected commute time between $a_1$ and $a_2$, where $m$ is the number of edges.

(c) By first-step analysis, compute $h(a_1 \to a_2)$ and verify that $h(a_1 \to a_2) + h(a_2 \to a_1) = C(a_1, a_2)$.