Strongly regular and strongly walk-regular graphs that admit perfect state transfer

TL;DR

This paper classifies strongly regular graphs with perfect state transfer and explores spectral conditions for strongly walk-regular graphs, combining algebraic number theory and spectral graph theory to advance understanding in quantum walk applications.

Contribution

It provides a complete classification of strongly regular graphs with perfect state transfer and characterizes spectral conditions for strongly walk-regular graphs.

Findings

Only $K_{2,2}$ and $K_{2,2,2}$ are strongly regular graphs with perfect state transfer.

Spectral form for strongly walk-regular graphs with perfect state transfer is identified.

Enumerated feasible spectra for such graphs up to $k=20$.

Abstract

We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph $K_{2,2}$ and the complete tripartite graph $K_{2,2,2}$. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form $\{[k]^1, [\frac{k}{2}]^{\alpha}, [0]^{\beta}, [-\frac{k}{2}]^{\gamma}\}$, and we enumerate all feasible spectra of this form up to $k=20$ with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory.