Average mixing matrix of trees

TL;DR

This paper studies the rank of the average mixing matrix in trees with distinct eigenvalues, establishing bounds and exploring cases where the rank approaches the upper limit, revealing structural properties of such matrices.

Contribution

It provides bounds on the rank of the average mixing matrix of trees and identifies families of trees with ranks significantly below the upper bound.

Findings

Rank of average mixing matrix is at most n/2 for trees with n vertices.

Most trees up to 20 vertices have ranks close to the upper bound.

An infinite family of trees with ranks bounded away from the upper bound is constructed.

Abstract

We investigate the rank of the average mixing matrix of trees, with all eigenvalues distinct. The rank of the average mixing matrix of a tree on $n$ vertices with $n$ distinct eigenvalues is upper-bounded by $\frac{n}{2}$. Computations on trees up to $20$ vertices suggest that the rank attains this upper bound most of the times. We give an infinite family of trees whose average mixing matrices have ranks which are bounded away from this upper bound. We also give a lower bound on the rank of the average mixing matrix of a tree.