Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs

TL;DR

This paper investigates the periodicity and perfect state transfer properties of Grover walks on quadratic unitary Cayley graphs, identifying all periodic cases and the specific values of n for perfect state transfer.

Contribution

It characterizes all periodic quadratic unitary Cayley graphs and determines the conditions for perfect state transfer, including infinitely many both integral and non-integral graphs.

Findings

All periodic quadratic unitary Cayley graphs are characterized.

Conditions for perfect state transfer are explicitly determined.

Infinitely many graphs exhibit periodicity, both integral and non-integral.

Abstract

The quadratic unitary Cayley graph $\mathcal{G}_{\mathbb{Z}_n}$ has vertex set $\mathbb{Z}_n: =\{0,1, \ldots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if and only if $u - v$ or $v-u$ is a square of some units in $\mathbb{Z}_n$. This paper explores the periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs. We determine all periodic quadratic unitary Cayley graphs. From our results, it follows that there are infinitely many integral as well as non-integral graphs that are periodic. Additionally, we also determine the values of $n$ for which the quadratic unitary Cayley graph $\mathcal{G}_{\mathbb{Z}_n}$ exhibits perfect state transfer.