Grover walks on unitary Cayley graphs and integral regular graphs

TL;DR

This paper investigates the properties of Grover walks on unitary Cayley graphs, characterizing their periodicity and conditions for perfect state transfer, and extends the analysis to integral regular graphs.

Contribution

It provides a complete characterization of periodicity and perfect state transfer for Grover walks on unitary Cayley graphs and integral regular graphs, identifying only four graphs with perfect state transfer.

Findings

Only four unitary Cayley graphs exhibit perfect state transfer.

Periodic Grover walks are characterized for integral regular graphs.

A necessary condition links periodicity to perfect state transfer in vertex-transitive graphs.

Abstract

The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if $\gcd(u - v, n) = 1$. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.