Random Walks and the Best Meeting Time for Trees

We consider random walks on a tree $G=(V,E)$ with stationary distribution ${\pi }_v=\mathrm{d}\mathrm{e}\mathrm{g}(v)\mathrm{/}2\mathrm{\mid }E\mathrm{\mid }$ 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 $T_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{e}\mathrm{t}}(G)={\min }_{w\in V}{\sum }_{v\in V}{\pi }_{v}H(v,w)$ 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.