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

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 $i\in{1,...,s}$. We characterize periodicity and perfect state transfer on $\mathcal{G}_R$ under the condition that $|R_i|/|M_i|\equiv 1 \pmod 4$ for $i\in{1,...,s}$. 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 $i\in{1,..., s}$, where $R_i$ is a local ring with maximal ideal $M_i$ for $i\in{0,...,s}$.