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 2017
TL;DR: This paper presents a meta-analysis of graph similarity testing for image processing and understanding in relation to software design and software development.
Abstract: IMAGE PROCESSING AND UNDERSTANDING BASED ON GRAPH SIMILARITY TESTING: ALGORITHM DESIGN AND SOFTWARE DEVELOPMENT
3 citations
11 Mar 2019
TL;DR: In this article, the authors introduced the notion of a submodular transformation F : {0, 1}n → Rm, which applies m sub-modular functions to the n-dimensional input vector.
Abstract: The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations.In this paper, we introduce the notion of a submodular transformation F : {0, 1}n → Rm, which applies m submodular functions to the n-dimensional input vector, and then introduce the notions of its Laplacian and normalized Laplacian. With these notions, we unify and generalize the existing Cheeger inequalities by showing a Cheeger inequality for submodular transformations, which relates the conductance of a submodular transformation and the smallest non-trivial eigenvalue of its normalized Laplacian. This result recovers the Cheeger inequalities for undirected graphs, directed graphs, and hypergraphs, and derives novel Cheeger inequalities for mutual information and directed information.Computing the smallest non-trivial eigenvalue of a normalized Laplacian of a submodular transformation is NP-hard under the small set expansion hypothesis. In this paper, we present a polynomial-time O(log n)-approximation algorithm for the symmetric case, which is tight, and a polynomial-time O(log2n + log n · log m)-approximation algorithm for the general case.We expect the algebra concerned with submodular transformations, or submodular algebra, to be useful in the future not only for generalizing spectral graph theory but also for analyzing other problems that involve piecewise linear transformations, e.g., deep learning.
3 citations
Posted Content•
TL;DR: This work derives an approach to directly compute the partial correlations based on concepts from inverse problem theory, which leads to new insights on open problems such as model selection and data preprocessing, as well as new approaches which relate the above application areas.
Abstract: While sparse inverse covariance matrices are very popular for modeling network connectivity, the value of the dense solution is often overlooked. In fact the L2-regularized solution has deep connections to a number of important applications to spectral graph theory, dimensionality reduction, and uncertainty quantification. We derive an approach to directly compute the partial correlations based on concepts from inverse problem theory. This approach also leads to new insights on open problems such as model selection and data preprocessing, as well as new approaches which relate the above application areas.
3 citations
Posted Content•
TL;DR: In this paper, the authors generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case when it is non-Hermitian.
Abstract: We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace the Hermiticity condition. We prove new variational estimates of the Rayleigh quotient for non-Hermitian matrices. As an application, a new Delsarte-Hoffman-type bound on the size of the largest independent set in a directed graph is developed. Our techniques consist in quantifying the impact of breaking the Hermitian symmetry of a matrix and are broadly applicable.
3 citations
27 Apr 2017
TL;DR: In this article, the energy of a graph is defined as the sum of the values of the eigenvalues of the adjacency matrix of the graph, which is defined in the context of spectral graph theory.
Abstract: In mathematics, the energy of a graph is the sum of the values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory. In this paper the concepts of non-commuting graph of dihedral groups are presented and the general formula for the energy of this associated graph is found.
3 citations