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

Koushik Bhakta; Bikash Bhattacharjya

doi:10.1016/j.disc.2026.115151

paper

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

arXiv:2502.10217 · doi:10.1016/j.disc.2026.115151

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 $G_{R}$ 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 $R_{1} \times \dots \times R_{s}$ , where each $R_{i}$ is a local ring with maximal ideal $M_{i}$ for $i \in {1, \dots, s}$ . We characterize periodicity and perfect state transfer on $G_{R}$ under the condition that $∣ R_{i} ∣/∣ M_{i} ∣ \equiv 1 (mod 4)$ for $i \in {1, \dots, s}$ . Also, we characterize periodicity and perfect state transfer on $G_{R}$ , where $R$ can be expressed as $R_{0} \times \dots \times R_{s}$ such that $∣ R_{0} ∣/∣ M_{0} ∣ \equiv 3 (mod 4)$ , and $∣ R_{i} ∣/∣ M_{i} ∣ \equiv 1 (mod 4)$ for $i \in {1, \dots, s}$ , where $R_{i}$ is a local ring with maximal ideal $M_{i}$ for $i \in {0, \dots, s}$ .

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

Abstract

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