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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2108.12864 |
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| _version_ | 1866911773073866752 |
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| author | Chakraborti, Debsoumya Kim, Jaehoon Kim, Jinha Kim, Minki Liu, Hong |
| author_facet | Chakraborti, Debsoumya Kim, Jaehoon Kim, Jinha Kim, Minki Liu, Hong |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_12864 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Well-mixing vertices and almost expanders Chakraborti, Debsoumya Kim, Jaehoon Kim, Jinha Kim, Minki Liu, Hong Combinatorics Discrete Mathematics 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). |
| title | Well-mixing vertices and almost expanders |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2108.12864 |