Quantum Walks on Regular Graphs and Eigenvalues

TL;DR

This paper investigates the spectral properties of matrices derived from quantum walks on strongly regular graphs, aiming to understand their potential to distinguish such graphs based on eigenvalues.

Contribution

It provides explicit eigenvalues for matrices related to quantum walks on regular graphs, extending previous work on spectral graph invariants.

Findings

Eigenvalues of S^+(U) and S^+(U^2) for regular graphs are derived.

The spectral approach may help distinguish strongly regular graphs.

Results support the use of quantum walk spectra in graph classification.

Abstract

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We find the eigenvalues of $S^+(U)$ and $S^+(U^2)$ for regular graphs.