Journal ArticleDOI
Review of Spectral Graph Theory: by Fan R. K. Chung
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TLDR
This book defines the Laplacian of a graph, a matrix closely related to the adjacency matrix, in analogy with the continuous case and studies the eigenvalues of this Laplace, which are related to many other more "discrete" invariants.Abstract:
Specifying a graph is equivalent to specifying its adjacency relation, which may be encoded in the form of a matrix. This suggests that study of the adjacency matrix from a linear-algebraic point of view might yield valuable information about graphs. In particular, any invariant associated to the matrix is also an invariant associated to the graph, and might have combinatorial meaning. Spectral graph theory is the study of the relationship between a graph and the eigenvalues of matrices (such as the adjacency matrix) naturally associated to that graph. This book looks at the subject from a geometric point of view, exploiting an analogy between a graph and a Riemannian manifold: Chung defines the Laplacian of a graph, a matrix closely related to the adjacency matrix, in analogy with the continuous case and studies the eigenvalues of this Laplacian.There are several reasons that these eigenvalues may be of interest. On the purely mathematical level, the eigenvalues have the advantage of being an extremely natural invariant which behaves nicely under operations such as Cartesian product and disjoint union. From a combinatorial point of view, the eigenvalues of a graph are related to many other more "discrete" invariants. From a geometric point of view, there are many respects in which the eigenvalues of a graph behave like the spectrum of a compact Riemannian manfiold. For the computationally-minded, the eigenvalues of a graph are easy to compute, and their relationship to other invariants can often yields good approximations to less tractible computations.read more
Citations
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Graph embedding as a new approach for unknown malware detection
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The journey of graph kernels through two decades
TL;DR: This survey will explore various kernels that operate on graph representations from its first appearance to discussion of current state of the art techniques in practice.
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A Nonlinear Orthogonal Non-Negative Matrix Factorization Approach to Subspace Clustering
TL;DR: A nonlinear NMF with explicit orthogonality and general kernel-based orthogonal multiplicative update rules to solve the subspace clustering problem and introduces a graph regularization to obtain a factorization that respects a local geometric structure of the data after the nonlinear mapping.
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Investigating the impact of feature selection on the prediction of solar radiation in different locations in Saudi Arabia
TL;DR: The results showed the importance of using feature selection methods in order to obtain a reliable prediction of the amount of solar radiation compared with using all the features available.
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T-ReX: a graph-based filament detection method
TL;DR: In this article, the authors propose a new approach for the automatic retrieval of the underlying filamentary structure from a 2D or 3D galaxy distribution using graph theory and the assumption that paths that link galaxies together with the minimum total length highlight the underlying distribution.
References
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Book
Spectral Graph Theory
TL;DR: Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigen values and quasi-randomness
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The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
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The complexity of computing the permanent
TL;DR: It is shown that the permanent function of (0, 1)-matrices is a complete problem for the class of counting problems associated with nondeterministic polynomial time computations.
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Random generation of combinatorial structures from a uniform
TL;DR: It is shown that exactly uniform generation of ‘efficiently verifiable’ combinatorial structures is reducible to approximate counting (and hence, is within the third level of the polynomial hierarchy).