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arXiv · 2108.12864

Well-mixing vertices and almost expanders

Debsoumya Chakraborti·Jaehoon Kim·Jinha Kim·Minki Kim·Hong Liu
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Abstract

We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time).

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Debsoumya Chakraborti, Jaehoon Kim, Jinha Kim, Minki Kim, Hong Liu. 2021-08-29. Well-mixing vertices and almost expanders. https://doi.org/10.1090/proc/16090

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