Diagonal entries of the average mixing matrix

TL;DR

This paper investigates the properties of the diagonal entries of the average mixing matrix in continuous quantum walks, focusing on extremal values and specific graph classes with constant diagonals.

Contribution

It classifies graphs with maximum trace of the average mixing matrix and provides constructions for graphs with constant diagonal entries.

Findings

Identified graphs with maximum and minimum trace of the average mixing matrix.

Classified graphs with constant diagonal entries in their average mixing matrices.

Provided explicit constructions for certain classes of graphs.

Abstract

We study the diagonal entries of the average mixing matrix of continuous quantum walks. The average mixing matrix is a graph invariant; it is the sum of the Schur squares of spectral idempotents of the Hamiltonian. It is non-negative, doubly stochastic and positive semi-definite. We investigate the diagonal entries of this matrix. We study the graphs for which the trace of the average mixing matrix is maximum or minimum and we classify those which are maximum. We give two constructions of graphs whose average mixing matrices have constant diagonal.