发表机构
School of Mathematics and Statistics, Qinghai Minzu University; Qinghai Institute of Applied Mathematics（青海民族大学数学与统计学院; 青海省应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分类了无权混合图上完美态转移的完备类及其归一化转移时间，给出了容许集合的充要条件，构造了任意有限大小的完备类，并刻画了定向图中的同时实现。
AI 中文摘要
对于无权混合图上的完美态转移（PST），我们对完备PST类的归一化转移时间进行了分类。一个包含零的有限集合$\Lambda\subset\mathbb R/\mathbb Z$在某个非平稳周期顶点处出现，当且仅当对所有$x,y\in\Lambda$，$\cos(2\pi(x-y))\in\mathbb Q$。每个容许集合都有一个连通的定向实现。我们还分类了在最小顶点周期处定向实现的可能返回相位。在公共最小顶点周期的有理数倍处的转移将完备类划分为大小至多为六的集合，或在返回相位为$-1$的定向图中大小至多为三的集合；这两个界都是紧的。我们构造了每个有限大小的完备类，包括所有不同顶点之间的转移发生在周期的无理数倍处且没有切换自同构将类顶点映射到不同类顶点的类。我们还刻画了具有指定的相对最小顶点周期和返回相位的连通定向图中的同时实现。证明结合了虚二次域限制与选择完备目标集的无权构造。
英文摘要
For perfect state transfer (PST) on unweighted mixed graphs, we classify the normalized transfer times of complete PST classes. A finite set $Λ\subset\mathbb R/\mathbb Z$ containing zero occurs at a nonstationary periodic vertex if and only if $\cos(2π(x-y))\in\mathbb Q$ for all $x,y\inΛ$. Every admissible set has a connected oriented realization. We also classify the possible return phases of oriented realizations at the minimum vertex period. Transfers at rational multiples of the common minimum vertex period partition a complete class into sets of size at most six, or at most three in an oriented graph with return phase $-1$; both bounds are sharp. We construct complete classes of every finite size, including classes in which all transfers between distinct vertices occur at irrational multiples of the period and no switching automorphism maps a class vertex to a distinct class vertex. We also characterize simultaneous realization in connected oriented graphs with prescribed relative minimum vertex periods and return phases. The proof combines an imaginary quadratic field restriction with an unweighted construction that selects the complete target set.
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