AI 中文总结
本文研究量子游走在笛卡尔积、联图和图补运算下的行为，刻画峰值态转移和相当好的态转移保持的条件，并给出图与其补图间量子游走差异的界。
AI 中文摘要
设 $U_X(t)$ 为图 $X$ 上相对于其邻接矩阵 $A$ 或拉普拉斯矩阵 $L$ 的量子游走的转移矩阵。本文研究了量子游走在笛卡尔积、联图和图补运算下的行为。我们有两个主要目标。首先，我们刻画了在这些运算下保持峰值态转移和相当好的态转移的条件，从而使我们能够构造具有这些性质的新图族。我们的第二个目标是分析图上的量子游走与其补图上的量子游走之间的关系。我们给出了 $f_{u,v}(t)=\big|U_{X^c}(t)_{u,v}-e^{it\delta}U_X(-t)_{u,v}\big|$ 和 $g_{u,v}(t)=\big||U_X(t)_{u,v}|-|U_{X^c}(t)_{u,v}|\big|$ 的界，其中当处理 $A$ 时 $\delta=-1$，否则 $\delta=n$。注意，$f_{u,v}(t)$ 和 $g_{u,v}(t)$ 都衡量了图与其补图中顶点 $u$ 和 $v$ 之间量子态转移行为的差异。如果 $X$ 是正则图或 $M=L$，则 $f_{u,v}(t)$ 的上界为 $\frac{2}{|V(X)|}$。如果 $X$ 是非正则的且 $M=A$，我们利用图的主特征值来获得 $f_{u,v}(t)$ 的一个仅依赖于 $A$ 的上界。我们还使用图的界定矩阵来给出 Nordhaus-Gaddum 型关系 $|U_X(t)_{u,v}|+|U_{X^c}(t)_{u,v}|$ 和 $|U_X(t)_{u,v}|\cdot |U_{X^c}(t)_{u,v}|$ 的界。最后，我们证明了对于某些图族，我们的大部分界是紧的。
英文摘要
Let $U_X(t)$ be the transition matrix of a quantum walk on a graph $X$ relative to its adjacency matrix $A$ or the Laplacian matrix $L$. This paper investigates the behavior of quantum walks under Cartesian products, joins, and graph complements. We have two main goals. First, we characterize the conditions such that peak state transfer and pretty good state transfer are preserved under these operations, allowing us to construct new families of graphs admitting these properties. Our second goal is to analyze the relationship between the quantum walks on a graph and its complement. We provide bounds for $f_{u,v}(t)=\big|U_{X^c}(t)_{u,v}-e^{itδ}U_X(-t)_{u,v}\big|$ and $g_{u,v}(t)=\big||U_X(t)_{u,v}|-|U_{X^c}(t)_{u,v}|\big|$, where $δ=-1$ when dealing with $A$ and $δ=n$ otherwise. Note that $f_{u,v}(t)$ and $g_{u,v}(t)$ both measure the difference between the behavior of quantum state transfer between vertices $u$ and $v$ in a graph and its complement. If $X$ is regular or $M=L$, then $f_{u,v}(t)$ is bounded above by $\frac{2}{|V(X)|}$. If $X$ is non-regular and $M=A$, then we utilize the main eigenvalues of a graph to obtain an upper bound for $f_{u,v}(t)$ which depends only on $A$. We also use the bounding matrix of the graph to give bounds for the Nordhaus-Gaddum type relations $|U_X(t)_{u,v}|+|U_{X^c}(t)_{u,v}|$ and $|U_X(t)_{u,v}|\cdot |U_{X^c}(t)_{u,v}|$. Finally, we demonstrate that most of our bounds are sharp for certain families of graphs.
Comments25 pages, 2 figures
