On the concentration of eigenvalues of random symmetric matrices

Alon, Noga; Krivelevich, Michael; Vu, Van

doi:10.1007/bf02785860

Cited by 111 publications

(181 citation statements)

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“…Fig. 2 suggests that in a constant fraction of the graphs in G n, 1 2 the eigenvector corresponding to µ n has three nodal domains.…”

Section: Question 1 How Many Nodal Domains (Strong or Weak) Do The Ementioning

confidence: 97%

“…Figure 2: The probability in G(n, 1 2 ) that the last eigenvector of the Laplacian has three nodal domains. For each n there were 500 experiments carried out.…”

Section: Question 1 How Many Nodal Domains (Strong or Weak) Do The Ementioning

confidence: 99%

“…Füredi and Komlós prove in [9, §3.3] (1)). As before, the theorem is derived by using a tail bound from [1].…”

Section: Theorem 23 (Tail Bound For Eigenvalues Of G(n P))mentioning

confidence: 99%

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Eigenvectors of Random Graphs: Nodal Domains

Dekel

Lee

Linial

2007

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that almost surely there are just two nodal domains and no exceptional vertices.

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“…Fig. 2 suggests that in a constant fraction of the graphs in G n, 1 2 the eigenvector corresponding to µ n has three nodal domains.…”

Section: Question 1 How Many Nodal Domains (Strong or Weak) Do The Ementioning

confidence: 97%

“…Figure 2: The probability in G(n, 1 2 ) that the last eigenvector of the Laplacian has three nodal domains. For each n there were 500 experiments carried out.…”

Section: Question 1 How Many Nodal Domains (Strong or Weak) Do The Ementioning

confidence: 99%

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Eigenvectors of Random Graphs: Nodal Domains

Dekel

Lee

Linial

2007

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…As an application of Theorem 8 we consider the concentration of eigenvalues of random symmetric matrices as analysed in [1]. Here we take k = [ 1; 1] and n = m+1 2 .…”

Section: Eigenvalues Of Random Symmetric Matricesmentioning

confidence: 99%

“…To prove our result we plagiarise the argument in [1], which is modi…ed only slightly to make Theorem 8 applicable.…”

Section: Theorem 10mentioning

confidence: 99%