## Introduction

Let $A$ be the adjacency matrix of a graph, with the spectral decomposition
\[A = \sum_r \theta_r E_r.\]
The continuous quantum walk is
\[U(t) = \exp(itA) = \sum_r e^{i\theta_r t} E_r.\]
The probability distributions of the walk are given by the *mixing matrix*
\[M(t) = U(t) \circ U(-t).\]
In general this does not converge. However, the average over time
\[\frac{1}{T} \int_0^T M(t) dt\]
does converge to the *average mixing matrix*
\[\widehat{M} = \sum_r E_r \circ E_r.\]
The average mixing matrix is symmetric and doubly stochastic, so each column is a probability distribution. We compute the entropy of each column and sum them up. This sum measures how far $\widehat{M}$ is from being flat.

The following is our data in .sobj format on trees and connected graphs.

- The entropy sum for trees on up to 14 vertices. data
- The entropy sum for connected graph on up to 8 vertices. data

There are graphs with the same entropy sum. Here are some examples.