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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TL;DR: In this paper, the spectral theory of 1-Laplacian on graphs and its applications to the Cheeger cut, max-cut, and multi-cut problems are surveyed, and the structure of eigenspace, nodal domains, multiplicities of eigenvalues and algorithms for graph cuts are collected.
Abstract: This is primarily an expository paper surveying up-to-date known results on the spectral theory of 1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.
13 citations
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TL;DR: A new canonical form as well as its relation with four structural models often encountered in practice and their corresponding graphs are presented and the block diagonalization of this form, which is performed using three Kronecker products and unsymmetric matrices, is studied.
Abstract: There are numerous applications of graph theory and algebraic graph theory in combinatorial optimization and optimal structural analysis. In this paper, a new canonical form as well as its relation with four structural models often encountered in practice and their corresponding graphs are presented. Furthermore, the block diagonalization of this form, which is performed using three Kronecker products and unsymmetric matrices, is studied. This block diagonalization leads to an efficient method for the eigensolution of adjacency and Laplacian matrices of special graphs. The eigenvalues and eigenvectors are used for efficient nodal ordering and partitioning of large structural models. The present method is far more simple than any existing general approach.
12 citations
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01 Jan 1995TL;DR: Spectral graph theory has a long history in representation theory and number theory and has been very useful for examining the spectra of strongly regular graphs with symmetries as mentioned in this paper, however, spectral graph theory is mostly algebraic.
Abstract: The study of eigenvalues of graphs has a long history. Since the early days, representation theory and number theory have been very useful for examining the spectra of strongly regular graphs with symmetries. In contrast, recent developments in spectral graph theory concern the effectiveness of eigenvalues in studying general (unstructured) graphs. The concepts and techniques, in large part, use essentially geometric methods.(Still, extremal and explicit constructions are mostly algebraic [20].) There has been a significant increase in the interaction between spectral graph theory and many areas of mathematics as well as other disciplines, such as physics, chemistry, communication theory, and computer sciences.
12 citations
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TL;DR: Extended Quantum Cuts is proposed, which consistently achieves an exquisite performance over all benchmark saliency detection datasets, containing around 18 k images in total and outperforms the state-of-the-art on a recently announced RGB-Depth saliency dataset.
Abstract: In this manuscript, an unsupervised salient object extraction algorithm is proposed for RGB and RGB-Depth images. Saliency estimation is formulated as a foreground detection problem. To this end, Quantum-Cuts (QCUT), a recently proposed spectral foreground detection method is investigated and extended to formulate the saliency estimation problem more efficiently. The contributions of this work are as follows: (1) a new proof for QCUT from spectral graph theory point of view is provided, (2) a detailed analysis of QCUT and comparison to well-known graph clustering methods are conducted, (3) QCUT is utilized in a multiresolution framework, (4) a novel affinity matrix construction scheme is proposed for better encoding of saliency cues into the graph representation and (5) a multispectral analysis for a richer set of salient object proposals is investigated. With the above improvements, we propose Extended Quantum Cuts, which consistently achieves an exquisite performance over all benchmark saliency detection datasets, containing around 18 k images in total. Finally, the proposed approach also outperforms the state-of-the-art on a recently announced RGB-Depth saliency dataset.
12 citations
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30 Aug 2010TL;DR: This work provides image segmentation based on Normalized Cut, with image representation based on the Component Tree and on its scale-space analysis, and presents a comparison between other image representations, as pixel grids, including multiscale graph decomposition formulation, and Watershed Transform.
Abstract: Graph partitioning, or graph cut, has been studied by several authors as a way of image segmenting. In the last years, the Normalized Cut has been widely used in order to implement graph partitioning, based on the graph spectra analysis (eigenvalues and eigenvectors). This area is known as Spectral Graph Theory. This work uses a hierarchical structure in order to represent images, the Component Tree. We provide image segmentation based on Normalized Cut, with image representation based on the Component Tree and on its scale-space analysis. Experimental results present a comparison between other image representations, as pixel grids, including multiscale graph decomposition formulation, and Watershed Transform. As the results show, the proposed approach, applied to different images, presents satisfying image segmentation.
12 citations