Topic
Spectral graph theory
About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.
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01 Jan 2015
TL;DR: In this article, a star-shaped affine regular polyhedra with a prescribed combinatorial structure was shown to be a regular polygon with affine equivalent to a regular one.
Abstract: For an arbitrary polygon generate a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buff on (1707–1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine
B
-regular polyhedra. The main result is the proof of existence of star-shaped affine $$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.
01 Jan 2021
TL;DR: In this article, the authors show how domination parameters have appeared in spectral graph theory, including the domination number γ, the total domination number, and the signed domination number (SVD).
Abstract: Spectral graph theory studies graphs through the eigenvalues and eigenvectors of matrices associated with them. In this chapter we show how domination parameters have appeared in spectral graph theory, including the domination number γ, the total domination number γt, and the signed domination number γs.
TL;DR: In this article , the authors considered the problem of the existence of a switching equivalent signed graph with Laplacian eigenvalues that are all main and determined all the switched equivalent signed graphs with this spectral property.
Abstract: For a graph G we consider the problem of the existence of a switching equivalent signed graph with Laplacian eigenvalues that are all main and the problem of determination of all switching equivalent signed graphs with this spectral property. Using a computer search we confirm that apart from K2 every connected graph with at most 7 vertices switches to at least one signed graph with the required property. This fails to hold for exactly 22 connected graphs with 8 vertices. If G is a cograph without repeated eigenvalues, then we give an iterative solution for the latter problem and the complete solution in the particular case when G is a threshold graph. The first problem is resolved positively for a particular class of chain graphs. The obtained results are applicable in control theory for generating controllable signed graphs based on Laplacian dynamics.
01 Jan 2016
TL;DR: This work addresses the problem of optimizing the decoding of JPEG-compressed images, employing an approach based on the “generalized” graph Laplacian, a higher-order generalization of the usual graph La placian.
Abstract: We address the problem of optimizing the decoding of JPEG-compressed images, employing an approach based on the “generalized” graph Laplacian, a higher-order generalization of the usual graph Laplacian. The optimal decoding problem is formulated as a non-smooth but convex problem over a graph, and solved via the alternating directions method of multipliers. While similar, graph-based optimized decoding techniques exist, the use of our generalized graph Laplacian enables better recovery of the original smooth image from the transform coefficients, especially those coded at low bit-rates. Experimental results highlight the performance of our proposed approach in terms of reconstructed distortion and visual quality. Comparisons with the method based on the usual graph Laplacian and other state-of-the-art methods are also given.
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TL;DR: Experimental results show that the proposed method outperforms frame-by-frame denoising from state-of-the-art static point cloud Denoising approaches.
Abstract: The prevalence of accessible depth sensing and 3D laser scanning techniques has enabled the convenient acquisition of 3D dynamic point clouds, which provide efficient representation of arbitrarily-shaped objects in motion. Nevertheless, dynamic point clouds are often perturbed by noise due to hardware, software or other causes. While many methods have been proposed for the denoising of static point clouds, dynamic point cloud denoising has not been studied in the literature yet. Hence, we address this problem based on the proposed spatio-temporal graph modeling, exploiting both the intra-frame similarity and inter-frame consistency. Specifically, we first represent a point cloud sequence on graphs and model it via spatio-temporal Gaussian Markov Random Fields on defined patches. Then for each target patch, we pose a Maximum a Posteriori estimation, and propose the corresponding likelihood and prior functions via spectral graph theory, leveraging its similar patches within the same frame and corresponding patch in the previous frame. This leads to our problem formulation, which jointly optimizes the underlying dynamic point cloud and spatio-temporal graph. Finally, we propose an efficient algorithm for patch construction, similar/corresponding patch search, intra- and inter-frame graph construction, and the optimization of our problem formulation via alternating minimization. Experimental results show that the proposed method outperforms frame-by-frame denoising from state-of-the-art static point cloud denoising approaches.