论文标题
复杂的hadamard对角线图
Complex Hadamard Diagonalisable Graphs
论文作者
论文摘要
鉴于最近对Hadamard可对角图的兴趣（Hadamard矩阵可对角线的图形），我们将此概念从真实的Hadamard矩阵中推广。我们提供了一些构建此类图的基本属性和方法。我们表明，一大批复杂的Hadamard可对角线图具有形成公平分区的顶点集，而Laplacian特征值甚至必须是整数。我们提供了许多复杂的Hadamard对角线图的示例和构造，包括两个特殊的图形类别：$ \ Mathbb {z} _r^d $上的Cayley图形以及非完整的扩展$ P $ - SUM（NEPS）。我们讨论了$（α，β）$的必要条件 - laplacian的分数复兴和完美的状态转移，以复杂的Hadamard可对角线图描述的连续时间量子步道，并提供了这种量子状态转移的示例。
In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex Hadamard matrices. We give some basic properties and methods of constructing such graphs. We show that a large class of complex Hadamard diagonalisable graphs have vertex sets forming an equitable partition, and that the Laplacian eigenvalues must be even integers. We provide a number of examples and constructions of complex Hadamard diagonalisable graphs, including two special classes of graphs: the Cayley graphs over $\mathbb{Z}_r^d$, and the non--complete extended $p$--sum (NEPS). We discuss necessary and sufficient conditions for $(α, β)$--Laplacian fractional revival and perfect state transfer on continuous--time quantum walks described by complex Hadamard diagonalisable graphs and provide examples of such quantum state transfer.
