正规凯莱图上格罗弗游走中的完美状态转移
Perfect state transfer in Grover walks on normal Cayley graphs
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中文总结 AI 辅助
研究正规凯莱图上格罗弗游走的完美状态转移，建立充要条件，推导阿贝尔群等上凯莱图的谱准则及组合特征，给出酉凯莱图完美状态转移完整特征，证明其中恰有四个图展示该转移。
中文摘要 AI 辅助
有限群$\Gamma$上的凯莱图$\operatorname{Cay}(\Gamma,S)$，若其连接集$S$是$\Gamma$的一些共轭类的并集，则称其为正规的。本文研究正规凯莱图上格罗弗游走中的完美状态转移。格罗弗游走是一种被广泛研究的离散时间量子游走。我们建立了正规凯莱图上发生完美状态转移的充要条件。作为应用，我们推导了阿贝尔群、双循环群和二面体群上凯莱图完美状态转移的明确谱准则。这些结果给出了几个展示完美状态转移的无限族凯莱图。我们进一步得到了二面体群和双循环群上凯莱图完美状态转移存在性的简单组合特征。我们的一般特征还恢复了一些先前已知的特殊情况结果。此外，我们得到了酉凯莱图上完美状态转移的完整特征。特别地，我们证明了酉凯莱图类中恰好有四个图展示完美状态转移。
英文摘要
A Cayley graph $\operatorname{Cay}(Γ,S)$ over a finite group $Γ$ is said to be normal if its connection set $S$ is a union of some conjugacy classes of $Γ$. This paper investigates perfect state transfer in Grover walks on normal Cayley graphs. The Grover walk is a widely studied discrete-time quantum walk. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on normal Cayley graphs. As applications, we derive explicit spectral criteria for perfect state transfer on Cayley graphs over abelian groups, dicyclic groups, and dihedral groups. These results yield several infinite families of Cayley graphs exhibiting perfect state transfer. We further obtain simple combinatorial characterizations of the existence of perfect state transfer on Cayley graphs over dihedral and dicyclic groups. Our general characterization also recovers a number of previously known results as special cases. As a further consequence, we obtain a complete characterization of perfect state transfer on unitary Cayley graphs. In particular, we prove that exactly four graphs in the class of unitary Cayley graphs exhibit perfect state transfer.
