Periodicity of Grover walks on distance-regular graphs

TL;DR

This paper investigates the conditions under which Grover walks are periodic on distance-regular graphs, extending known results from strongly regular graphs and providing new necessary conditions for periodicity.

Contribution

It introduces new necessary conditions for periodic Grover walks on distance-regular graphs and applies these to re-establish results for strongly regular graphs.

Findings

Identified classes of distance-regular graphs with periodic Grover walks

Derived necessary conditions for periodicity on general distance-regular graphs

Provided a new proof for periodicity in strongly regular graphs

Abstract

Characterizations graphs of some classes to induce periodic Grover walks have been studied for recent years. In particular, for the strongly regular graphs, it has been known that there are only three kinds of such graphs. Here, we focus on the periodicity of the Grover walks on distance-regular graphs. The distance-regular graph can be regarded as a kind of generalization of the strongly regular graphs and the typical graph with an equitable partition. In this paper, we find some classes of such distance-regular graphs and obtain some useful necessary conditions to induce periodic Grover walks on the general distance-regular graphs. Also, we apply this necessary condition to give another proof for the strong regular graphs.