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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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17 Jun 2012
TL;DR: It is shown that the most straightforward version of a higher-dimensional Cheeger inequality fails: for every k>1, there is an infinite family of k-dimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.
Abstract: Eigenvalues associated to graphs are a well-studied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n,p) are known quite precisely. We consider generalizations of these matrices to simplicial complexes of higher dimensions and study their eigenvalues for the Linial--Meshulam model Xk(n,p) of random k-dimensional simplicial complexes on n vertices. We show that for p=Ω(log n/n), the eigenvalues of both, the higher-dimensional adjacency matrix and the Laplacian, are a.a.s.~sharply concentrated around two values.In a second part of the paper, we discuss a possible higher-dimensional analogue of the Discrete Cheeger Inequality. This fundamental inequality expresses a close relationship between the eigenvalues of a graph and its combinatorial expansion properties; in particular, spectral expansion (a large eigenvalue gap) implies edge expansion.Recently, a higher-dimensional analogue of edge expansion for simplicial complexes was introduced by Gromov, and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is natural to ask whether there is a higher-dimensional version of Cheeger's inequality. We show that the most straightforward version of a higher-dimensional Cheeger inequality fails: for every k>1, there is an infinite family of k-dimensional complexes that are spectrally expanding (there is a large eigenvalue gap for the Laplacian) but not combinatorially expanding.
60 citations
TL;DR: In this article, the spectral radius of a non-regular graph was investigated and the ratio of any two components of a maximal eigenvector of a graph corresponding to its spectral radius was investigated.
60 citations
TL;DR: A normalized digraph Laplacian Γ is introduced and defined for digraphs, and it is proved that its Moore–Penrose pseudoinverse is the discrete Green’s function of the Di Placian matrix as an operator on digraph's, and that it is the normalized fundamental matrix of the Markov chain governing random walks onDigraphs.
Abstract: In this paper we extend and generalize the standard spectral graph theory (or random-walk theory) on undirected graphs to digraphs. In particular, we introduce and define a normalized digraph Laplacian (Diplacian for short) Γ for digraphs, and prove that (1) its Moore–Penrose pseudoinverse is the discrete Green’s function of the Diplacian matrix as an operator on digraphs, and (2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive a new formula for computing hitting and commute times in terms of the Moore–Penrose pseudoinverse of the Diplacian, or equivalently, the singular values and vectors of the Diplacian. Furthermore, we show that the Cheeger constant defined in [Chung 05] is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric, the largest singular value of the skewed Laplacian ∇=(Γ−Γ T )/2, to quantify and measure the degree of asymmetry in a digraph. Using this measure, we est...
60 citations
18 Dec 1997
TL;DR: In this article, the analysis of Hill's operator D 2 + q(x) for qeven and periodic is extended from the real line to homogeneous trees T. The spectrum is exactly described when the degree of the tree is greater than two, in which case there are both spectral bands and eigenvalues.
Abstract: The analysis of Hill’s operator D 2 + q(x)for qeven and periodic is extended from the real line to homogeneous trees T. Generalizing the classical problem, a detailed analysis of Hill’s equation and its related operatortheoryon L 2 (T)isprovided. Themultipliersforthisnewversion of Hill’s equation are identied and analyzed. An explicit description of theresolventis given. The spectrumis exactly describedwhen thedegree of the tree is greater than two, in which case there are both spectral bands and eigenvalues. Spectral projections are computed by means of an eigenfunction expansion. Long time asymptotic expansions for the associated semigroup kernel are also described. A summation formula expresses the resolvent for a regular graph as a function of the resolvent of its covering homogeneous tree and the covering map. In the case of a nite regular graph, a trace formula relates the spectrum of the Hill’s operator to the lengths of closed paths in the graph.
59 citations
TL;DR: A GA Laplacian matrix is defined which determines the geometric-arithmetic index of a graph and satisfies properties similar to the ones of the classical LaplACian matrix.
59 citations