The maximum relaxation time of a random walk

TL;DR

This paper determines the asymptotic minimum spectral gap of the normalized Laplacian for all simple connected graphs, identifies the extremal graph, and applies this to establish sharp bounds on the relaxation time of random walks, confirming a conjecture.

Contribution

It provides the asymptotic minimum spectral gap and identifies the extremal double kite graph, leading to sharp bounds on relaxation times and improving eigenvalue-diameter inequalities.

Findings

Minimum spectral gap is asymptotically (54/n^3)

Double kite graph asymptotically achieves this minimum

Sharp upper bounds for relaxation time of random walk

Abstract

We show the minimum spectral gap of the normalized Laplacian over all simple, connected graphs on $n$ vertices is $(1+o(1))\tfrac{54}{n^3}$. This minimum is achieved asymptotically by a double kite graph. Consequently, this leads to sharp upper bounds for the maximum relaxation time of a random walk, settling a conjecture of Aldous and Fill. We also improve an eigenvalue-diameter inequality by giving a new lower bound for the spectral gap of the normalized Laplacian. This eigenvalue lower bound is asymptotically best possible.