Complex Hadamard Diagonalisable Graphs

• URL: http://arxiv.org/abs/2001.00251v2
• Date: Sun, 19 Jul 2020 19:22:05 GMT
• Title: Complex Hadamard Diagonalisable Graphs
• Authors: Ada Chan, Shaun Fallat, Steve Kirkland, Jephian C.-H. Lin, Shahla Nasserasr, and Sarah Plosker
• Abstract summary: We show that a large class of complex Hadamard diagonalisable graphs have sets forming an equitable partition. We provide examples and constructions of complex Hadamard diagonalisable graphs. We discuss necessary and sufficient conditions for $(alpha, beta)$--Laplacian fractional revival and perfect state transfer.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $(\alpha, \beta)$--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.

Related papers

• The exact quantum chromatic number of Hadamard graphs [0.0]
  We compute the quantum chromatic numbers of Hadamard graphs of order $n=2N$ for $N$ a multiple of $4$. We also compute the exact quantum chromatic number of the categorical product of Hadamard graphs.
  arXiv  Detail & Related papers  (2024-09-27T14:28:57Z)
• A generalization of quantum pair state transfer [0.0]
  An $s$-pair state in a graph is a quantum state of the form $mathbfe_u+smathbfe_v$. We develop the theory of perfect $s$-pair state transfer in continuous quantum walks.
  arXiv  Detail & Related papers  (2024-04-25T14:45:49Z)
• Graph Generation via Spectral Diffusion [51.60814773299899]
  We present GRASP, a novel graph generative model based on 1) the spectral decomposition of the graph Laplacian matrix and 2) a diffusion process. Specifically, we propose to use a denoising model to sample eigenvectors and eigenvalues from which we can reconstruct the graph Laplacian and adjacency matrix. Our permutation invariant model can also handle node features by concatenating them to the eigenvectors of each node.
  arXiv  Detail & Related papers  (2024-02-29T09:26:46Z)
• Weakly Hadamard diagonalizable graphs and Quantum State Transfer [0.0]
  We study Hadamard diagonalizable graphs in the context of quantum state transfer. We provide numerous properties and constructions of weak Hadamard matrices and weakly Hadamard diagonalizable graphs.
  arXiv  Detail & Related papers  (2023-07-04T18:04:47Z)
• Implications of sparsity and high triangle density for graph representation learning [67.98498239263549]
  Recent work has shown that sparse graphs containing many triangles cannot be reproduced using a finite-dimensional representation of the nodes. Here, we show that such graphs can be reproduced using an infinite-dimensional inner product model, where the node representations lie on a low-dimensional manifold.
  arXiv  Detail & Related papers  (2022-10-27T09:15:15Z)
• Strong Cospectrality and Twin Vertices in Weighted Graphs [0.0]
  We show that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to arbitrary Hermitian matrices. We also generalize known results about equitable and almost equitable partitions, and use these to determine which joins of the form $Xvee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.
  arXiv  Detail & Related papers  (2021-11-01T21:18:42Z)
• Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields [51.07460861448716]
  This paper presents a convex-analytic framework to learn from data. We show that a triangular convexity decomposition is guaranteed by a transform of the corresponding to its upper part.
  arXiv  Detail & Related papers  (2021-09-17T17:46:12Z)
• Non-PSD Matrix Sketching with Applications to Regression and Optimization [56.730993511802865]
  We present dimensionality reduction methods for non-PSD and square-roots" matrices. We show how these techniques can be used for multiple downstream tasks.
  arXiv  Detail & Related papers  (2021-06-16T04:07:48Z)
• High-Dimensional Gaussian Process Inference with Derivatives [90.8033626920884]
  We show that in the low-data regime $ND$, the Gram matrix can be decomposed in a manner that reduces the cost of inference to $mathcalO(N2D + (N2)3)$. We demonstrate this potential in a variety of tasks relevant for machine learning, such as optimization and Hamiltonian Monte Carlo with predictive gradients.
  arXiv  Detail & Related papers  (2021-02-15T13:24:41Z)
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs [62.997667081978825]
  We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. We show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.
  arXiv  Detail & Related papers  (2020-08-11T17:58:13Z)
• Cospectrality preserving graph modifications and eigenvector properties via walk equivalence of vertices [0.0]
  Cospectrality is a powerful generalization of exchange symmetry and can be applied to all real-valued symmetric matrices. We show that the powers of a matrix with cospectral vertices induce further local relations on its eigenvectors. Our work paves the way for flexibly exploiting hidden structural symmetries in the design of generic complex network-like systems.
  arXiv  Detail & Related papers  (2020-07-15T10:54:31Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.