Further results on the perfect state transfer in integral circulant graphs
Further results on the perfect state transfer in integral circulant graphs
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DOI：
10.1016/j.camwa.2010.11.005
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Petković;Milan Basic
中科院分区：
文献类型：
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作者：
M. Petković;Milan Basic
对于给定的图G，用A表示它的邻接矩阵，F（t）=exp（iAt）。我们说G中存在完美态转移（PST），如果|F（τ）ab| =1，对于某些顶点a，B和一个正的真实的数τ.这一性质对于最近邻耦合量子自旋网络的建模是非常重要的。研究了整数循环图（具有整数特征值的循环图）中完美状态转移的存在性。Saxena等人已经获得了关于这个主题的一些结果。（2007）[5]，Bašić et al.（2009）[6]和Basić & Petković（2009）[7]。本文证明了存在一个n阶整循环图具有完美状态转移当且仅当4 <$n。已经发现了几类整数循环图，它们对于可被4整除的n值具有完美的状态转移。此外，我们证明了不存在的PST的其他几类整循环图的阶是可除的4。这些类涵盖了一类图的除数集正好包含两个元素。所得结果部分回答了哪些整数循环图具有完美状态转移这一主要问题。
For a given graph G, denote by A its adjacency matrix and F(t)=exp(iAt). We say that there exist a perfect state transfer (PST) in G if |F(τ)ab|=1, for some vertices a,b and a positive real number τ. Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) [5], Bašić et al. (2009) [6] and Basić & Petković (2009) [7]. In this paper, we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4∣n. Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4. Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by 4. These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer.