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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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Posted ContentDOI
TL;DR: This paper develops a framework in which a stochastic process based on a set of interacting random walks on a graph is constructed and it is shown that a suitably scaled version of the stochastics process converges to the Fiedler vector for a sufficiently large number of walks.
Abstract: The Fiedler vector of a graph, namely the eigenvector corresponding to the second smallest eigenvalue of a graph Laplacian matrix, plays an important role in spectral graph theory with applications in problems such as graph bi-partitioning and envelope reduction. Algorithms designed to estimate this quantity usually rely on a priori knowledge of the entire graph, and employ techniques such as graph sparsification and power iterations, which have obvious shortcomings in cases where the graph is unknown, or changing dynamically. In this paper, we develop a framework in which we construct a stochastic process based on a set of interacting random walks on a graph and show that a suitably scaled version of our stochastic process converges to the Fiedler vector for a sufficiently large number of walks. Like other techniques based on exploratory random walks and on-the-fly computations, such as Markov Chain Monte Carlo (MCMC), our algorithm overcomes challenges typically faced by power iteration based approaches. But, unlike any existing random walk based method such as MCMCs where the focus is on the leading eigenvector, our framework with interacting random walks converges to the Fiedler vector (second eigenvector). We also provide numerical results to confirm our theoretical findings on different graphs, and show that our algorithm performs well over a wide range of parameters and the number of random walks. Simulations results over time varying dynamic graphs are also provided to show the efficacy of our random walk based technique in such settings. As an important contribution, we extend our results and show that our framework is applicable for approximating not just the Fiedler vector of graph Laplacians, but also the second eigenvector of any time reversible Markov Chain kernel via interacting random walks.

3 citations

Book ChapterDOI
22 Nov 2004
TL;DR: In this paper, the spectral curve of the Lax representation becomes the graph of integrable systems, such as the open Toda molecule, and generalizations for which a function lives on a cylinder, torus or a Riemann surface of higher genus.
Abstract: For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a function $A(\lambda)$. Those integrable systems provide an interesting ``toy model'' of separation of variables. Examples of this type of integrable systems are presented along with generalizations for which $A(\lambda)$ lives on a cylinder, a torus or a Riemann surface of higher genus.

3 citations

Posted Content
TL;DR: Generalization of the tensor concepts to non-euclidean domain, orders of magnitude speed-up, low-memory requirement and significantly enhanced performance at low SNR are the key aspects of this framework.
Abstract: We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building on this novel decomposition we construct a general class of convex optimization problems for approximately solving low-rank tensor inverse problems, such as tensor Robust PCA. The whole framework is named as 'Multilinear Low-rank tensors on Graphs (MLRTG)'. Our theoretical analysis shows: 1) MLRTG stands on the notion of approximate stationarity of multi-dimensional signals on graphs and 2) the approximation error depends on the eigen gaps of the graphs. We demonstrate applications for a wide variety of 4 artificial and 12 real tensor datasets, such as EEG, FMRI, BCI, surveillance videos and hyperspectral images. Generalization of the tensor concepts to non-euclidean domain, orders of magnitude speed-up, low-memory requirement and significantly enhanced performance at low SNR are the key aspects of our framework.

3 citations

01 Jan 2008
TL;DR: In this article, the Laplacian polynomial of a graph is expressed in terms of the characteristic polynomials of the induced subgraphs of the graph.
Abstract: In this paper, we express the Laplacian polynomial of a graph in terms of the characteristic polynomials of its induced subgraphs. Further the Laplacian polynomial of a regular graph is expressed in terms of derivatives of its characteristic polynomial. In the sequel we obtain the Laplacian polynomial of a complement of a graph in terms of the characteristic polynomial of induced subgraphs of a graph. Using these we obtain the number of spanning trees of a graph.

3 citations

Journal ArticleDOI
TL;DR: This paper calculates the Laplacian energy of some grid based networks using the multiset of eigenvalues of LaPLacian matrix.
Abstract: of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called its spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, perfect matching etc. Laplacian spectrum is the multiset of eigenvalues of Laplacian matrix. The Laplacian energy of a graph is the sum of the absolute values of its Laplacian eigenvalues. In this paper we calculate the Laplacian energy of some grid based networks

3 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981