The Maximum Commute Time of Regular Graphs

For a connected simple regular graph $G$, let $\tau (G)$ be the maximum commute time for simple random walk, and let $t_n$ be the maximum of $\tau (G)$ over all such graphs of order $n$. Aldous and Fill conjectured that $t_n\sim 3n^2\mathrm{/}2$. We determine $t_n$ exactly and classify all maximizers for every even $n\ge 10$ and every odd $n\ge 15$; both thresholds are best possible. The maximizers are cubic block chains joined by bridges at even orders and quartic block chains joined at cut vertices at odd orders. The exact formulas specify the linear corrections according to the order modulo four and five, respectively, and imply $t_n\sim \{\begin{array}{ll}{\displaystyle \frac{3}{2}}{n}^2,& n\text{ even},\\ n^2,& n\text{ odd},\end{array}n\to \mathrm{\infty }\mathrm{.}$ Thus the conjectured asymptotic formula holds along even orders but fails along odd orders. The proof combines unit-current potential orderings, cut-current inequalities, and sharp block resistance bounds, supplemented by finite computations in exact rational arithmetic.