arXiv subjects
Publications and source records attributed to Paul Meunier.
We study several noncommutative properties of 0-hyperbolic graphs. In particular, we prove that 0-hyperbolicity is preserved under quantum isomorphism. We also compute the quantum automorphism groups of 0-hyperbolic graphs and characterise the ones with quantum symmetry.
We prove that for every pair of quantum isomorphic graphs, their block trees and their block graphs are isomorphic, and that such an isomorphism can be chosen so that the corresponding blocks are quantum isomorphic -- in particular, 2-connectedness is preserved under quantum isomorphism. We conclude with some corollaries, including obtaining some necessary conditions on a pair of quantum isomorphic, not isomorphic graphs with a minimal number of vertices.
We prove that the bunkbed conjecture is preserved under gluing along a vertex. As a consequence, every minimal counterexample is 2-connected. More generally, for any class of graphs closed under taking 2-connected components, the study of the bunkbed conjecture reduces to the case of 2-connected graphs in that class. In particular, the conjecture holds for forests and block graphs.
We initiate a systematic study of quantum properties of finite graphs, namely, quantum asymmetry, quantum symmetry, and quantum isomorphism. We define the Schmidt alternative for a class of graphs, which reveals to be a useful tool for studying quantum symmetries of graphs. After showing that quantum isomorphic graphs have quantum isomorphic centers and connected components, we solve the aforementioned problems for the classes of cographs and forests. We also compute their quantum automorphism groups for the first time. In doing so, we extend to the noncommutative setting a theorem of Jordan. Using general results on $\mathcal F$-cographs, we extend the precedent results to $\mathcal G_5$-cographs and tree-cographs, two distinct strictly proper superclasses of cographs and forests respectively. Finally, we show that quantum isomorphic planar graphs are isomorphic.
Our motivating application is a real-world problem: COVID-19 classification from CT imaging, for which we present an explainable Deep Learning approach based on a semi-supervised classification pipeline that employs variational autoencoders to extract efficient feature embedding. We have optimized the architecture of two different networks for CT images: (i) a novel conditional variational autoencoder (CVAE) with a specific architecture that integrates the class labels inside the encoder layers and uses side information with shared attention layers for the encoder, which make the most of the contextual clues for representation learning, and (ii) a downstream convolutional neural network for supervised classification using the encoder structure of the CVAE. With the explainable classification results, the proposed diagnosis system is very effective for COVID-19 classification. Based on the promising results obtained qualitatively and quantitatively, we envisage a wide deployment of our developed technique in large-scale clinical studies.Code is available at https://git.etrovub.be/AVSP/ct-based-covid-19-diagnostic-tool.git.