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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Journal Article•
TL;DR: In this article, an inequality on eigenvalues of the Laplacian matrix of a graph G = V, E is presented, and applied to estimate the eigen values of the graph G.
1 citations
TL;DR: The generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree as discussed by the authors, and the eigenvalues of the Laplacian and adjacency matrices of the graph G are characterized in terms of their multiplicities.
01 Jan 2015
TL;DR: This paper proposes a measure of similarity for the different states of the Markov process and uses techniques from spectral graph theory to show that the perturbed structure can be obtained by looking at the spectrum of a graph defined on the proposed similarity matrix.
Abstract: Multiscale Markov processes are used to model and control stochastic dynamics across different scales in many applications areas such as electrical engineering, finance, and material science. A commonly used mathematical representation that captures multiscale stochastic dynamics is that of singularly perturbed Markov processes. Dimensionality reductions techniques for this class of stochastic optimal control problems have been studied for many years. However, it is typically assumed that the structure of perturbed process and its dynamics are known. In this paper, we show how to infer the structure of a singularly perturbed Markov process from data. We propose a measure of similarity for the different states of the Markov process and then use techniques from spectral graph theory to show that the perturbed structure can be obtained by looking at the spectrum of a graph defined on the proposed similarity matrix.
Posted Content•
TL;DR: In this article, it was shown that G[rK1 [sK2] is DQS under certain conditions, where r, s are natural numbers and K1 and K2 denote the complete graphs on one vertex and two vertices, respectively.
Abstract: In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G[rK1 [sK2 is DQS under certain conditions, where r, s are natural numbers and K1 and K2 denote the complete graphs on one vertex and two vertices, respectively. Applying these results, some DQS graphs with independent edges and isolated vertices are obtained
25 May 2015
TL;DR: In this article, it was shown that graphs whose translation operators exhibit semigroup behavior are those whose eigenvectors of the Laplacian form a Hadamard matrix.
Abstract: The authors of [1] have used spectral graph theory to define a Fourier transform on finite graphs. With this definition, one can use elementary properties of classical time-frequency analysis to define time-frequency operations on graphs including convolution, modulation, and translation. Many of these graph operators have properties that match our intuition in Euclidean space. The exception lies with the translation operator. In particular, translation does not form a group, i.e., T i T j = T i+j . We prove that graphs whose translation operators exhibit semigroup behavior are those whose eigenvectors of the Laplacian form a Hadamard matrix.