Cheeger‐like inequalities for the largest eigenvalue of the graph Laplace operator

Jost, Jürgen; Mulas, Raffaella

doi:10.1002/jgt.22664

Cited by 3 publications

(6 citation statements)

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“…Since we know, from Lemma 4.2, that max f RQ 1 (f ) = 1, this proves that the upper bound in (22) shrinks to an equality for p = 1.…”

Section: Signed Coloring Numbermentioning

confidence: 58%

“…The fact that the upper bound in (22) shrinks to an equality for p = 1 is particularly interesting because this is similar to what happens for the Cheeger constant h in the case of graphs, and for the Cheeger-like constant Q defined in [22] for graphs and generalized in [31] for hypergraphs. In fact, we have that:…”

Section: Remark 77mentioning

confidence: 70%

“…3. In (22) we again have something similar, because the quantity that bounds λ n from below for p equals λ n for 1 .…”

Section: Remark 77mentioning

confidence: 85%

“…Of course, the main difference between the last case and the first two is that h and Q are constants that are independent of p, while the quantity in (22) changes when p changes.…”

Section: Remark 77mentioning

confidence: 99%

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p-Laplace Operators for Oriented Hypergraphs

2021

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“…Since we know, from Lemma 4.2, that max f RQ 1 (f ) = 1, this proves that the upper bound in (22) shrinks to an equality for p = 1.…”

Section: Signed Coloring Numbermentioning

confidence: 58%

Section: Remark 77mentioning

confidence: 70%

“…3. In (22) we again have something similar, because the quantity that bounds λ n from below for p equals λ n for 1 .…”

Section: Remark 77mentioning

confidence: 85%

“…Of course, the main difference between the last case and the first two is that h and Q are constants that are independent of p, while the quantity in (22) changes when p changes.…”

Section: Remark 77mentioning

confidence: 99%

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p-Laplace Operators for Oriented Hypergraphs

2021

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“…Cheeger-like constant Some further recent results [41] can be also rediscovered via Lovász extension.…”

Section: Multiplicative Cheeger Constant For Instancementioning

confidence: 90%

Discrete-to-Continuous Extensions: Lovász extension, optimizations and eigenvalue problems

Jost¹,

Zhang²

2021

Preprint

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In this paper, we use various versions of Lovász extension to systematically derive continuous formulations of problems from discrete mathematics. This will take place in the following context:-For combinatorial optimization problems, we systematically develop equivalent continuous versions, thereby making tools from convex optimization, fractional programming and more general continuous algorithms like the stochastic subgradient method available for such optimization problems. Among other applications, we present an effective iteration scheme combining the inverse power and the steepest descent method to relax a Dinkelbach-type scheme for solving the equivalent continuous optimization. These results are natural and nontrivial generalizations of the related works by Hein's group [30][31][32]. -For some combinatorial quantities like Cheeger-type constants, we suggest a nonlinear eigenvalue problem for a pair of Lovász extensions of certain functions, which encodes certain combinatorial structures.This theory has several applications to quantitative and combinatorial problems, including(1) The equivalent continuous representations for the max k-cut problem, various Cheeger sets and isoperimetric constants are constructed. This also initiates a study of Dirichlet and Neumann 1-Laplacians on graphs, in which the nodal domain property and Cheeger-type equalities are presented. Among them, some Cheeger constants using different versions of vertex-boundary introduced in expander graph theory [8], are transformed into continuous forms, which reprove the inequalities and identities on graph Poincare profiles proposed by Hume [33][34][35].(2) Also, we derive a new equivalent continuous representation of the graph independence number, which can be compared with the Motzkin-Straus theorem. More importantly, an equivalent continuous optimization for the chromatic number is provided, which seems to be the first continuous representation of the graph vertex coloring number. We provide the first continuous reformulation of the frustration index in signed networks, and we find a connection to the so-called modularity measure. Graph matching numbers, submodular vertex covers and multiway partition problems can also be studied in our framework.

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Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology

Mulas

Horak²,

Jost

2022

Understanding Complex Systems

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Cheeger‐like inequalities for the largest eigenvalue of the graph Laplace operator

Abstract: We define a new Cheeger‐like constant for graphs and we use it for proving Cheeger‐like inequalities that bound the largest eigenvalue of the normalized Laplace operator.

Cited by 3 publications

References 13 publications

p-Laplace Operators for Oriented Hypergraphs

p-Laplace Operators for Oriented Hypergraphs

Discrete-to-Continuous Extensions: Lovász extension, optimizations and eigenvalue problems

Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology

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