A note on mixing times of planar random walks

TL;DR

This paper constructs a family of planar graphs with bounded degree demonstrating a specific lower bound on the spectral gap related to graph diameter, simplifying previous constructions and discussing bounds involving average distances.

Contribution

It provides a simplified construction of planar graphs with a lower bound on the spectral gap related to diameter, and discusses limitations involving average squared distances.

Findings

Constructed planar graphs with degree ≤ 5 satisfying the eigenvalue bound.

Showed that such bounds do not hold when replacing diameter with average squared distance.

Provided bounds relating spectral gap to average pairwise distances in planar graphs.

Abstract

We present an infinite family of finite planar graphs $\{X_n\}$ with degree at most five and such that for some constant $c > 0$, $$ \lambda_1(X_n) \geq c(\frac{\log \diam(X_n)}{\diam(X_n)})^2\,, $$ where $\lambda_1$ denotes the smallest non-zero eigenvalue of the graph Laplacian. This significantly simplifies a construction of Louder and Souto. We also remark that such a lower bound cannot hold when the diameter is replaced by the average squared distance: There exists a constant $c > 0$ such that for any family $\{X_n\}$ of planar graphs we have $$ \lambda_1(X_n) \leq c (\frac{1}{|X_n|^2} \sum_{x,y \in X_n} d(x,y)^2)^{-1}\,, $$ where $d$ denotes the path metric on $X_n$.