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
本文分类了实对称矩阵幂下完美态转移的指数条件，利用谱算术确定最小转移时间，并完整分类了超立方体、循环图、Johnson 图及路径的邻接幂。核心贡献是揭示了奇偶性与谱对称性对转移存在性的决定性作用。
AI 中文摘要
对于实对称矩阵 $H$ 和不同的顶点 $a,b$，我们分类了使得 $H^k$ 从 $a$ 到 $b$ 具有完美态转移（PST）的指数 $k$。如果它们支撑的特征值是某个共同正数的整数倍，那么每个奇数指数都归结为 $H$，每个正偶数指数都归结为 $H^2$。我们利用支撑谱差的最大公约数来确定最小转移时间。对于有理对称矩阵，源顶点支撑关于零的对称性意味着在无需可公度性假设的情况下具有奇数幂等价性；这包括所有二部图。如果源顶点支撑包含零，则在一个正偶数幂下具有 PST 意味着在所有正偶数幂下都具有 PST。对于对称的三点二次谱，其外投影符号一致且与中心符号不同，非零有理平移恰好留下一个 PST 指数。我们分类了超立方体、循环图和 Johnson 图的所有邻接幂，以及路径的所有邻接平方。特别地，$P_7$ 的邻接矩阵仅在指数 $2$ 时从顶点 $2$ 到顶点 $6$ 具有 PST。
英文摘要
For a real symmetric matrix $H$ and distinct vertices $a,b$, we classify exponents $k$ for which $H^k$ has perfect state transfer (PST) from $a$ to $b$. If their supported eigenvalues are integer multiples of a common positive number, every odd exponent reduces to $H$ and every positive even exponent reduces to $H^2$. We determine the minimum transfer times using a greatest common divisor of supported spectral differences. For rational symmetric matrices, symmetry of the source vertex support about zero implies the odd-power equivalence without a commensurability assumption; this includes all bipartite graphs. If the source vertex supports zero, PST under one positive even power implies PST under every positive even power. For a symmetric three-point quadratic spectrum whose outer projection signs agree and differ from the central sign, a nonzero rational shift leaves exactly one PST exponent. We classify all adjacency powers of hypercubes, cycles, and Johnson graphs, and all adjacency squares of paths. In particular, the adjacency matrix of $P_7$ has PST from vertex $2$ to vertex $6$ only at exponent $2$.
Comments24 pages, 0 figures
