We consider random walks on a tree $G=(V,E)$ with stationary distribution $\pi_v = \mathrm{ deg}(v)/2|E|$ for $v \in V$. Let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at vertex $v$ to reach vertex $w$. We characterize the extremal tree structures for the best meeting time $T_{ \mathrm{ bestmeet}}(G) = \min_{ w \in V} \sum_{ v \in V} \pi_v H(v,w)$ for trees of order $n$ with diameter $d$. The best meeting time is maximized by the balanced double broom graph, and it is minimized by the balanced lever graph.
Categories: math.CO math.PR