Pretty good state transfer on double stars
Pretty good state transfer on double stars
复制标题
双星上相当好的状态转移
DOI：
10.1016/j.laa.2012.10.006
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发表时间：
2012-06
影响因子：
1.1
通讯作者：
Chris Godsil
中科院分区：
文献类型：
--
作者：
Xiaoxia Fan;Chris Godsil
设A是图X的邻接矩阵，U（t）=exp（itA）.我们认为A作用在CxV（X）上，并将该空间的标准基取为V（X）中u的向量eu。量子物理学家说，如果存在标量γ使得U（τ）eu=γev，则在时间τ处我们有从顶点u到v的完美态转移。（由于U（t）是酉的，所以<$γ=1 <$。）例如，如果X是d-立方体，u和v的距离为d，那么我们在π/2时刻有从u到v的完美状态转移。尽管存在这个美好的家庭，但很明显，完美的状态转移是罕见的。因此，我们考虑一个松弛：我们说，如果有一个复数γ，并且对于每个正的真实的γ，都有一个时间t，使得τ U（t）eu-γev τ <τ，则我们有从u到v的相当好的状态转移。同样，我们必须|γ| =1时。在最近的一篇论文中，Godsil、柯克兰、Severini和Smith表明，我们在路径Pnif的端点之间具有相当好的状态转移，并且只有n+1是2的幂、素数或素数的两倍。（仅对于P2和P3，端顶点之间存在完美的状态转移。）令人惊讶的是，相当好的状态转移的发生是以数论条件为特征的。本文推广了相当好态转移理论。我们提供了什么是唯一的第二个家庭的图，其中相当不错的状态转移发生。我们使用的图是双星图Sk，我们证明了完美状态转移不会发生在这个家庭中的任何图。我们证明了当n>2时，在S2，n中与三阶顶点相邻的两个端点之间有很好的状态转移.当k，n>2时，我们证明了两个度至少为3的顶点之间不存在完美的状态转移，并且证明了当且仅当这两个顶点都是k+1且4k+1不是完美的正方形时，它们之间才有很好的状态转移.由此我们再次发现，完全态转移的存在依赖于一个数论条件。同样有趣的是，虽然没有双星有完美的状态转移，但有一些双星承认有相当好的状态转移。
Let A be the adjacency matrix of a graph X and suppose U(t)=exp(itA). We view A as acting on CxV(X)and take the standard basis of this space to be the vectors eufor u in V(X). hysicists say that we have perfect state transfer from vertex u to v at time τ if there is a scalar γ such that U(τ)eu=γev. (Since U(t) is unitary, ‖γ=1‖.) For example, if X is the d-cube and u and v are at distance d then we have perfect state transfer from u to v at time π/2. Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from u to v if there is a complex number γ and, for each positive real ϵ there is a time t such that ‖U(t)eu-γev‖<ϵ. Again we necessarily have |γ|=1. In a recent paper Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path Pnif and only n+1 is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for P2and P3.) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we extend the theory of pretty good state transfer. We provide what is only the second family of graphs where pretty good state transfer occurs. The graphs we use are the double-star graphs Sk,ℓ, these are trees with a vertex of degree k+1 adjacent to a vertex of degree ℓ+1, and all other vertices of degree one. We prove that perfect state transfer does not occur in any graph in this family. We show that if ℓ>2, then there is pretty good state transfer in S2,ℓbetween the two end vertices adjacent to the vertex of degree three. If k,ℓ>2, we prove that there is never perfect state transfer between the two vertices of degree at least three, and we show that there is pretty good state transfer between them if and only these vertices both have degree k+1 and 4k+1 is not a perfect square. Thus we find again the the existence of perfect state transfer depends on a number theoretic condition. It is also interesting that although no double stars have perfect state transfer, there are some that admit pretty good state transfer.
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影响因子：
2.9
作者：
L. Vinet;A. Zhedanov
通讯作者：
L. Vinet;A. Zhedanov
DOI：
10.1007/978-1-4613-0163-9
发表时间：
2001-04
期刊：
--
影响因子：
--
作者：
C. D. Godsil;G. Royle
通讯作者：
C. D. Godsil;G. Royle
影响因子：
2.9
作者：
Kay, Alastair
通讯作者：
Kay, Alastair
DOI：
10.1016/j.disc.2011.06.032
发表时间：
2011-02
期刊：
Discret. Math.
影响因子：
--
作者：
C. Godsil
通讯作者：
C. Godsil
影响因子：
2.7
作者：
Nitin Saxena;S. Severini;I. Shparlinski
通讯作者：
Nitin Saxena;S. Severini;I. Shparlinski