Open problems of Paul Erd�s in graph theory

Chung, Fan

doi:10.1002/(sici)1097-0118(199705)25:1<3::aid-jgt1>3.0.co;2-r

Cited by 57 publications

(73 citation statements)

References 115 publications

(184 reference statements)

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“…The fact of λ 1 > 0 follows from the connectivity of G. The eigenvalue λ 1 is intimately related to the rate of convergence of random walks. The reader is referred to [4] for numerous properties concerning eigenvalues of the normalized Laplacian. Although the combinatorial Laplacian is useful for various flow problems in the study of electrical networks, the spectrum of combinatorial Laplacian is not effective (except for almost regular graphs) for applications requiring isoperimetric properties.…”

Section: Laplacian the Green's Function And Pagerankmentioning

confidence: 99%

PageRank and Random Walks on Graphs

Chung

Zhao

2010

Bolyai Society Mathematical Studies

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Abstract. We examine the relationship between PageRank and several invariants occurring in the study of random walks and electrical networks. We consider a generalized version of hitting time and effective resistance with an additional parameter which controls the 'speed' of diffusion. We will establish their connection with PageRank . Through these connections, a combinatorial interpretation of PageRank is given in terms of rooted spanning forests by using a generalized version of the matrix-tree theorem. Using PageRank, we will illustrate that the generalized hitting time leads to finding sparse cuts and efficient approximation algorithms for PageRank can be used for approximating hitting time and effective resistance.

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Section: Laplacian the Green's Function And Pagerankmentioning

confidence: 99%

PageRank and Random Walks on Graphs

Chung

Zhao

2010

Bolyai Society Mathematical Studies

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“…Spectral Coordinates The decomposition of the Laplacian matrix L = X T ΛX reveals the graph spectrum [7] which comprises the eigenvalues Λ = diag(λ 0 , λ 1 , ..., λ |V | ) (in increasing order) and their associated eigenmodes X = X (0) , X (1) , ..., X (|V |) (a |V | × |V | matrix where columns X (·) are eigenmodes). The first eigenmode is trivial (λ 0 = 0) and the following non-trivial eigenmodes are the fundamental modes of vibrations of a shape depicted by I Ω .…”

Section: Spectral Correspondencementioning

confidence: 99%

“…However, as in most registration methods, transformation updates based on the image gradients are inherently limited by their local scope. Secondly, we introduce a new update scheme for groupwise registration based on the spectral decomposition of graph Laplacians [7,23,13], that is invariant to shape isometry and is capable of capturing large deformations during the construction of the atlas. We provide two forms of our groupwise registration framework that we name the Groupwise Log-Demons (GL-Demons, faster and suited for local nonrigid deformations), and the Groupwise Spectral Log-Demons (GSL-Demons, slower but capable of capturing very large deformations).…”

Section: Introductionmentioning

confidence: 99%

Groupwise Spectral Log-Demons Framework for Atlas Construction

Lombaert

Grady

Pennec

et al. 2013

Medical Computer Vision. Recognition Techniques and Applications in Medical Imaging

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Abstract. We introduce a new framework to construct atlases from images with very large and complex deformations. The atlas is build in parallel with groupwise registrations by extending the symmetric Log-Demons algorithm. We describe and evaluate two forms of our framework: the Groupwise LogDemons (GL-Demons) is faster but is limited to local nonrigid deformations, and the Groupwise Spectral Log-Demons (GSL-Demons) is slower but, due to isometry-invariant representations of images, can construct atlases of organs with high shape variability. We demonstrate our framework by constructing atlases from hearts with high shape variability.

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“…For the latter, graph spectral representations [10], which are invariant to isometry (preserving geodesic distances), can capture large and complex deformations. Spectral methods [10,11,12,13,14] are popular for general graph partitioning and are applied in computer vision for mesh matching [15,16,17,18] and shape retrieval [19]. Pioneered in the late 80s, [12,13,14], spectral correspondence methods match points in shapes by comparing eigenvectors of a proximity matrix derived from the mesh structure.…”

Section: Introductionmentioning

confidence: 99%

Spectral Demons – Image Registration via Global Spectral Correspondence

Lombaert

Grady

Pennec

et al. 2012

Lecture Notes in Computer Science

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Abstract. Image registration is a building block for many applications in computer vision and medical imaging. However the current methods are limited when large and highly non-local deformations are present. In this paper, we introduce a new direct feature matching technique for non-parametric image registration where efficient nearest-neighbor searches find global correspondences between intensity, spatial and geometric information. We exploit graph spectral representations that are invariant to isometry under complex deformations. Our direct feature matching technique is used within the established Demons framework for diffeomorphic image registration. Our method, called Spectral Demons, can capture very large, complex and highly non-local deformations between images. We evaluate the improvements of our method on 2D and 3D images and demonstrate substantial improvement over the conventional Demons algorithm for large deformations.

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Open problems of Paul Erd�s in graph theory

Cited by 57 publications

References 115 publications

PageRank and Random Walks on Graphs

PageRank and Random Walks on Graphs

Groupwise Spectral Log-Demons Framework for Atlas Construction

Spectral Demons – Image Registration via Global Spectral Correspondence

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