Well-mixing vertices and almost expanders

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).