Well-mixing vertices and almost expanders
Debsoumya Chakraborti, Jaehoon Kim, Jinha Kim, Minki Kim, Hong Liu

TL;DR
This paper proves that regular graphs with many well-mixing vertices are essentially expanders, have no small separators, contain long cycles, and that well-mixing from a positive fraction of vertices implies near-universal well-mixing.
Contribution
It establishes a strong link between local mixing properties and global expansion, answering a longstanding question and improving cycle length bounds in sparse graphs.
Findings
Graphs with many well-mixing vertices are virtually expanders.
Such graphs have no small separators.
Existence of long cycles in sparse regular graphs with many well-mixing vertices.
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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Taxonomy
TopicsStochastic processes and statistical mechanics · Limits and Structures in Graph Theory · Markov Chains and Monte Carlo Methods
