Well-mixing vertices and almost expanders
Combinatorics
2024-02-09 v4 Discrete Mathematics
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).
Cite
@article{arxiv.2108.12864,
title = {Well-mixing vertices and almost expanders},
author = {Debsoumya Chakraborti and Jaehoon Kim and Jinha Kim and Minki Kim and Hong Liu},
journal= {arXiv preprint arXiv:2108.12864},
year = {2024}
}
Comments
accepted in PAMS