AI 中文总结
该研究对有向循环图中不同顶点间的完美状态传递进行完整分类，证明其仅在导体Δ为3、4、8时发生，推导相关公式并确定连通阶与枚举结果。
AI 中文摘要
有向循环图上的连续时间量子行走由其埃尔米特邻接矩阵的傅里叶特征值决定。我们对每个非空有向循环图中不同顶点间的完美状态传递（PST）进行分类。我们证明，每个此类图由导体为Δ的奇本原二次狄利克雷特征、一组最大公约数类，以及对每个选定类的两种定向选择来描述。对于阶为n的图，我们推导了每个傅里叶特征值的显式公式，且不假设n/Δ与Δ互素。我们证明，PST仅在Δ∈{3,4,8}时发生，并针对每个导体给出了连接集的充分必要条件。等价地，存在PST的有向循环图的无平方因子根基恰好为1、2和3。更一般地，当λ_j=√Dη_j（其中η_j∈ℤ）时，整数η_j满足的同余式确定了所有PST对与时间、最小周期，以及支持多状态传递（MST）的最大顶点集。在该类中，良好状态传递等价于PST。我们还确定了连通阶并枚举了所得的图。
英文摘要
The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor $Δ$, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order $n$, we derive an explicit formula for every Fourier eigenvalue without assuming that $n/Δ$ is coprime to $Δ$. We prove that PST occurs only for $Δ\in\{3,4,8\}$ and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly $1$, $2$, and $3$. More generally, when $λ_j=\sqrt{D}η_j$ with $η_j\in\mathbb{Z}$, congruences satisfied by the integers $η_j$ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.
Comments31 pages
