Random Walks and the Best Meeting Time for Trees

TL;DR

This paper characterizes extremal tree structures that optimize the best meeting time for random walks, identifying the balanced double broom as maximizing and the balanced lever as minimizing this metric.

Contribution

It provides a complete characterization of trees that extremize the best meeting time for random walks, including explicit extremal structures.

Findings

Balanced double broom maximizes the best meeting time.

Balanced lever minimizes the best meeting time.

Results depend on tree diameter and structure.

Abstract

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.