The Exact Mixing Time for Trees with Fixed Diameter

TL;DR

This paper precisely determines the extremal structure of trees with fixed diameter that maximize the mixing time of random walks, identifying the balanced double broom as the unique maximizer.

Contribution

It characterizes the exact extremal trees for mixing time among trees with fixed diameter, identifying the balanced double broom as the unique maximizer.

Findings

Balanced double broom maximizes mixing time among trees with fixed diameter.

Exact extremal structure for mixing time on trees characterized.

Mixing time achieved uniquely by the balanced double broom.

Abstract

We characterize the extremal structure for the exact mixing time for random walks on trees $T_{n,d}$ of order $n$ with diameter $d$. Given a graph $G=(V,E)$, let $H(v,\pi)$ denote the expected length of an optimal stopping rule from vertex $v$ to the stationary distributon $\pi$. We show that the quantity $\max_{G \in T_{n,d} } T_{\mbox{mix}}(G) = \max_{G \in T_{n,d} } \max_{v \in V} H(v,\pi)$ is achieved uniquely by the balanced double broom.