The Laplacian Spectrum of a Graph II

Grone, Robert; Merris, Russell

doi:10.1137/s0895480191222653

Cited by 378 publications

(196 citation statements)

References 14 publications

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“…, which motivated a conjecture of Grone and Merris [19,Conjecture 1] that was proven by Grone [18]. Grone's result strengthens Proposition 8.1 to the following majorization inequality in the case k = 2:…”

Section: Some Easy Spectrum Boundssupporting

confidence: 52%

“…A simplicial complex K has associated to it chain groups C i (K) and boundary maps ∂ i : C i (K) → C i−1 (K) satisfying ∂ i+1 ∂ i = 0, which are used to define and compute Our second motivation was the hope that the spectra of shifted simplicial complexes might be extremal in some way that leads to inequalities for the spectra of all complexes. This led us to the following conjecture, generalizing a conjecture of Grone and Merris [19,Conjecture 2] for graphs. As with the above theorem, it is phrased in terms of ∂ i ∂ T i rather than L i , and for k-families rather than simplicial complexes.…”

Section: Introductionmentioning

confidence: 80%

“…The goal of this section is to recall and present evidence for Conjecture 1.2, which generalizes a conjecture of Grone and Merris [19,Conjecture 2].…”

Section: A Conjecture On the Spectramentioning

confidence: 85%

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Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs.We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.

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Section: Some Easy Spectrum Boundssupporting

confidence: 52%

Section: Introductionmentioning

confidence: 80%

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Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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“…For more on algebraic connectivity of trees and related results, see [1,2,8,[10][11][12]. A main goal of this paper is to introduce the notion of combinatorial Perron value, and investigate its properties.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Perron values of trees and bottleneck matrices

Andrade

Dahl

2016

Linear and Multilinear Algebra

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The algebraic connectivity a(G) of a graph G is an important parameter, defined as the second smallest eigenvalue of the Laplacian matrix of G. If T is a tree, a(T ) is closely related to the Perron values (spectral radius) of so-called bottleneck matrices of subtrees of T . In this setting we introduce a new parameter called the combinatorial Perron value ρc. This value is a lower bound on the Perron value of such subtrees; typically ρc is a good approximation to ρ. We compute exact values of ρc for certain special subtrees. Moreover, some results concerning ρc when the tree is modified are established, and it is shown that, among trees with given distance vector (from the root), ρc is maximized for caterpillars.

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“…In this paper we extend the ideas of these two proofs and find a generalization of Grone's result (4), and another lower bound on the sum of the largest Laplacian eigenvalues, which is closely related to a bound of Grone and Merris [4].…”

Section: Eigenvalue Interlacingmentioning

confidence: 66%

An interlacing approach for bounding the sum of Laplacian eigenvalues of graphs

Abiad

Fiol

Haemers

et al. 2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone & Merris. As a consequence we obtain inequalities involving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number. Eigenvalue interlacingThroughout this paper, G = (V, E) is a finite simple graph with n = |V | vertices. Theorem 1.1. Let A be a real symmetric n × n matrix with eigenvalues λ 1 ≥ · · · ≥ λ n . For some m < n, let S be a real n × m matrix with orthonormal columns, S ⊤ S = I, and consider the matrix B = S ⊤ AS, with eigenvalues µ 1 ≥ · · · ≥ µ m . Then, (a) the eigenvalues of B interlace those of A, that is,(b) if the interlacing is tight, that is, for some 0 ≤ k ≤ m, λ i = µ i , i = 1, . . . , k, and µ i = λ n−m+i , i = k + 1, . . . , m, then SB = AS.

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The Laplacian Spectrum of a Graph II

Cited by 378 publications

References 14 publications

Shifted simplicial complexes are Laplacian integral

Shifted simplicial complexes are Laplacian integral

Combinatorial Perron values of trees and bottleneck matrices

An interlacing approach for bounding the sum of Laplacian eigenvalues of graphs

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