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arXiv · 2510.24387

Random Walks and the Best Meeting Time for Trees

Andrew Beveridge·Ari Holcombe Pomerance
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Abstract

We consider random walks on a tree $G=(V,E)$ with stationary distribution $π_v = \mathrm{deg}(v)/2|E|$ 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 $T_{\mathrm{bestmeet}}(G) = \min_{w \in V} \sum_{v \in V} π_v H(v,w)$ 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.

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Andrew Beveridge, Ari Holcombe Pomerance. 2025-10-28. Random Walks and the Best Meeting Time for Trees. https://arxiv.org/abs/2510.24387

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