哪些加权循环网络具有完美的状态转移？
Which weighted circulant networks have perfect state transfer?
摘要：
基于加权图的量子自旋网络中的完美态转移存在性问题是近年来许多作者提出的问题。我们给出了一个简单的条件来表征加权循环图允许完美的状态转移的特征值。这是通过扩展由Saxena，Severini和Shparlinski获得的网络中量子周期性存在的结果和加权循环图中的积分图的特征来完成的。最后，我们找到了一类支持完美状态转移的加权循环图，这些图完全覆盖了无权情况下具有完美状态转移的循环图。事实上，我们证明了存在一个加权积分循环图的$n$顶点有完美的状态转移当且仅当$n$是偶数。此外，我们还证明了其他几类偶数阶的加权整循环图不存在完美状态转移。
英文摘要：
The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini and Shparlinski and characterizing integral graphs among weighted circulant graphs. Finally, classes of weighted circulant graphs supporting perfect state transfer are found. These classes completely cover the class of circulant graphs having perfect state transfer in the unweighted case. In fact, we show that there exists an weighted integral circulant graph with $n$ vertices having perfect state transfer if and only if $n$ is even. Moreover we prove the non-existence of perfect state transfer for several other classes of weighted integral circulant graphs of even order.
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