A Combinatorial Characterization of Constant Mixing Time

TL;DR

This paper provides a combinatorial characterization of graphs with constant mixing time, expanding understanding beyond spectral methods by introducing a new small-set bipartite density condition.

Contribution

It introduces a novel combinatorial criterion for constant mixing time, bridging spectral and expansion-based characterizations.

Findings

Characterizes graphs with constant mixing time using a new combinatorial condition.

Shows this condition is weaker than spectral radius constraints.

Demonstrates the condition is stronger than small-set vertex expansion.

Abstract

Classical spectral graph theory characterizes graphs with logarithmic mixing time. In this work, we present a combinatorial characterization of graphs with constant mixing time. The combinatorial characterization is based on the small-set bipartite density condition, which is weaker than having near-optimal spectral radius and is stronger than having near-optimal small-set vertex expansion.