Quantum isomorphism of graphs from association schemes

• URL: http://arxiv.org/abs/2209.04581v2
• Date: Wed, 26 Oct 2022 16:30:28 GMT
• Title: Quantum isomorphism of graphs from association schemes
• Authors: Ada Chan and William J. Martin
• Abstract summary: We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result of Man\v{c}inska and Roberson shows that graphs $G$ and $H$ are quantum isomorphic if and only if, for any planar graph $F$, the number of graph homomorphisms from $F$ to $G$ is equal to the number of graph homomorphisms from $F$ to $H$. A generalization of partition functions called "scaffolds" affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.

Related papers

• Detecting Homeomorphic 3-manifolds via Graph Neural Networks [0.0]
  We study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers.
  arXiv  Detail & Related papers  (2024-09-01T12:58:09Z)
• NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability [0.18749305679160366]
  We show that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. This homomorphism indistinguishability characterization also allows us to give a randomized-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.
  arXiv  Detail & Related papers  (2024-07-15T11:48:09Z)
• 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)
• Finding the Missing-half: Graph Complementary Learning for Homophily-prone and Heterophily-prone Graphs [48.79929516665371]
  Graphs with homophily-prone edges tend to connect nodes with the same class. Heterophily-prone edges tend to build relationships between nodes with different classes. Existing GNNs only take the original graph during training.
  arXiv  Detail & Related papers  (2023-06-13T08:06:10Z)
• Planar #CSP Equality Corresponds to Quantum Isomorphism -- A Holant Viewpoint [0.0]
  Graph homomorphism is the special case where each of $mathcalF$ and $mathcalF'$ contains a single symmetric 0-1-valued binary constraint function. We show that any pair of sets $mathcalF$ and $mathcalF'$ of real-valued, arbitrary-arity constraint functions give the same value on any planar #CSP instance.
  arXiv  Detail & Related papers  (2022-12-06T21:38:40Z)
• Casting graph isomorphism as a point set registration problem using a simplex embedding and sampling [0.0]
  A graph can be represented as a point set in enough dimensions using a simplex embedding and sampling. The isomorphism of them corresponds to the existence of a perfect registration between the point set forms of the graphs. The related idea of equivalence classes suggests that graph canonization may be an important tool in tackling graph isomorphism problem.
  arXiv  Detail & Related papers  (2021-11-15T12:16:21Z)
• AnchorGAE: General Data Clustering via $O(n)$ Bipartite Graph Convolution [79.44066256794187]
  We show how to convert a non-graph dataset into a graph by introducing the generative graph model, which is used to build graph convolution networks (GCNs) A bipartite graph constructed by anchors is updated dynamically to exploit the high-level information behind data. We theoretically prove that the simple update will lead to degeneration and a specific strategy is accordingly designed.
  arXiv  Detail & Related papers  (2021-11-12T07:08:13Z)
• 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)
• Factorizable Graph Convolutional Networks [90.59836684458905]
  We introduce a novel graph convolutional network (GCN) that explicitly disentangles intertwined relations encoded in a graph. FactorGCN takes a simple graph as input, and disentangles it into several factorized graphs. We evaluate the proposed FactorGCN both qualitatively and quantitatively on the synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2020-10-12T03:01:40Z)
• 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)
• Graph Homomorphism Convolution [19.94778871237204]
  We show that graph homomorphism numbers provide a natural invariant (isomorphism invariant and $mathcalF$-invariant) embedding maps which can be used for graph classification. By choosing $mathcalF$ whose elements have bounded tree-width, we show that the homomorphism method is efficient compared with other methods.
  arXiv  Detail & Related papers  (2020-05-03T23:56:20Z)

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.