Random Walks and the Best Meeting Time for Trees
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917048009883648 |
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| author | Beveridge, Andrew Pomerance, Ari Holcombe |
| author_facet | Beveridge, Andrew Pomerance, Ari Holcombe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Walks and the Best Meeting Time for Trees Beveridge, Andrew Pomerance, Ari Holcombe Combinatorics Probability 05C81 (primary), 05C05 (secondary), 60J10 (secondary) 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. |
| title | Random Walks and the Best Meeting Time for Trees |
| topic | Combinatorics Probability 05C81 (primary), 05C05 (secondary), 60J10 (secondary) |
| url | https://arxiv.org/abs/2510.24387 |