在Grover游动中允许完全状态转移的价为4的循环图

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2025-04-29 DOI:10.1016/j.jcta.2025.106064

Sho Kubota , Kiyoto Yoshino

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引用次数: 0

摘要

我们完全刻画了允许完全状态转移的价为4的循环图。那些价3的就不承认了。另一方面，价为4的循环图只在两个无限族中允许完全状态转移：一个是詹发现的，另一个是新发现的，而其他的都不允许。推导这些结果的主要工具是图的对称性和特征值。我们描述了基于图的对称性的完美状态转移发生的必要条件，图的对称性在数学上是指图的自同构。对于特征值，如果存在完全状态转移，则相应各向同性随机游走的某些特征值必须是代数整数的一半。考虑到这一点，我们利用已知的结果环的整数场。

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Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks

We completely characterize circulant graphs with valency up to 4 that admit perfect state transfer. Those of valency 3 do not admit it. On the other hand, circulant graphs with valency 4 admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.

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来源期刊

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.