State transfer in Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings

TL;DR

This paper investigates the conditions for periodicity and perfect state transfer of Grover walks on specific Cayley graphs over finite commutative rings, providing complete characterizations for various ring structures.

Contribution

It offers necessary and sufficient conditions for periodicity and perfect state transfer on unitary and quadratic unitary Cayley graphs over finite commutative rings, extending previous understanding.

Findings

Characterized when Cayley graphs exhibit periodicity.

Determined rings with perfect state transfer.

Provided conditions based on ring decompositions and properties.

Abstract

This paper focuses on periodicity and perfect state transfer of Grover walks on two well-known families of Cayley graphs, namely, the unitary Cayley graphs and the quadratic unitary Cayley graphs. Let $R$ be a finite commutative ring. The unitary Cayley graph $G_R$ has vertex set $R$, where two vertices $u$ and $v$ are adjacent if $u-v$ is a unit in $R$. We provide a necessary and sufficient condition for the periodicity of the Cayley graph $G_R$. We also completely determine the rings $R$ for which $G_R$ exhibits perfect state transfer. The quadratic unitary Cayley graph $\mathcal{G}_R$ has vertex set $R$, where two vertices $u$ and $v$ are adjacent if $u-v$ or $v-u$ is a square of some units in $R$. It is well known that any finite commutative ring $R$ can be expressed as $R_1\times\cdots\times R_s$, where each $R_i$ is a local ring with maximal ideal $M_i$ for $i\in\{1,\ldots,s\}$. We characterize periodicity and perfect state transfer on $\mathcal{G}_R$ under the condition that $|R_i|/|M_i|\equiv 1 \pmod 4$ for $i\in\{1,\ldots,s\}$. Also, we characterize periodicity and perfect state transfer on $\mathcal{G}_R$, where $R$ can be expressed as $R_0\times\cdots\times R_s$ such that $|R_0|/|M_0|\equiv3\pmod 4$, and $|R_i|/|M_i|\equiv1\pmod4$ for $i\in\{1,\ldots, s\}$, where $R_i$ is a local ring with maximal ideal $M_i$ for $i\in\{0,\ldots,s\}$.