arXiv 0806.20742008-06math.COquant-ph

周期图

Periodic Graphs

Chris Godsil

备注：: 19 pages, 1 figure. Fixes errors and typos

摘要：

设$X$是$n$顶点上的图，具有邻接矩阵$A$，并设$H(t)$表示矩阵值函数$\exp(iAt)$。如果$u$和$v$是$X$中不同的顶点，我们说从$u$到$v$的完美状态转移}发生，如果存在时间$τ$使得$|H(τ)_{u,v}|=1$。如果$u\in V(X)$和存在时间$\sg$使得$|H(\sg)_{u,u}|=1$，我们说$X$在$u$是周期性的，周期为$\sg$。我们表明，如果从$u$到$v$的完美状态转移发生在时间$τ$，则$X$在$u$和$v$都是周期的，周期为$2τ$。我们推广了以前的工作，证明了一个正则图，至少有四个不同的特征值是周期的关于某些顶点的当且仅当其特征值是整数。我们证明了，对于一类图$X$，包括所有的点传递图，如果完美的状态转移发生在时间$τ$，那么$H(τ)$是一个标量倍的置换矩阵的二阶没有不动点。利用一定的Hadamard矩阵，我们构造了一个新的无限图族，其上发生完美状态转移。

英文摘要：

Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer}from $u$ to $v$ occurs if there is a time $τ$ such that $|H(τ)_{u,v}|=1$. If $u\in V(X)$ and there is a time $\sg$ such that $|H(\sg)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\sg$. We show that if perfect state transfer from $u$ to $v$ occurs at time $τ$, then $X$ is periodic at both $u$ and $v$ with period $2τ$. We extend previous work by showing that a regular graph with at least four distinct eigenvalues is periodic with respect to some vertex if and only if its eigenvalues are integers. We show that, for a class of graphs $X$ including all vertex-transitive graphs, if perfect state transfer occurs at time $τ$, then $H(τ)$ is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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