Expander Graphs — Both Local and Global

Chapman, Michael; Linial, Nathan; Peled, Yuval

doi:10.1007/s00493-019-4127-8

Cited by 10 publications

(10 citation statements)

References 20 publications

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“…In this subsection, let thus P denote the order-5 4-simplex honeycomb, the automorphism group of which is the Coxeter group W with diagram 5 , sometimes known as H 5 . This example already answers the question asked in [CLP20] positively.…”

Section: Regular Polytope Psupporting

confidence: 54%

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Constructing highly regular expanders from hyperbolic Coxeter groups

Conder¹,

Lubotzky²,

Schillewaert³

et al. 2020

Preprint

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A graph X is defined inductively to be (a0, . . . , an−1)-regular if X is a0-regular and for every vertex v of X, the sphere of radius 1 around v is an (a1, . . . , an−1)-regular graph. Such a graph X is said to be highly regular (HR) of level n if an−1 = 0. Chapman, Linial and Peled [CLP20] studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist.In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system (W, S) and a subset M of S, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope PW,M , which form an infinite family of expander graphs when (W, S) is indefinite and PW,M has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of (W, S). The expansion stems from applying superapproximation to the congruence subgroups of the linear group W .This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked in [CLP20].

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Section: Regular Polytope Psupporting

confidence: 54%

“…Motivated by questions related to PCP-theory, Chapman, Linial and Peled [CLP20] initiated a systematic study of HR-graphs of level 2, that is, (a, b)-regular graphs. They were mainly interested in such graphs which are expanders "globally and locally".…”

Section: Introductionmentioning

confidence: 99%

Constructing highly regular expanders from hyperbolic Coxeter groups

Conder¹,

Lubotzky²,

Schillewaert³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…. , a n )-regular" graphs) following some recent work on expander graphs and PCP theory, in [3] followed by [4]. It is a very natural and basic question, and yet does not seem to be trivial and not much seems to be known about it.…”

Section: Introductionmentioning

confidence: 99%

“…It is a very natural and basic question, and yet does not seem to be trivial and not much seems to be known about it. In fact, the only contribution to this topic prior to [3] that we are aware of was by Zelinka [7], who pointed out some basic necessary conditions for existence of (k, t)-regular graphs, constructed a large range of examples using some basic graph products, and then ruled out the existence of (7,4)-regular graphs in a somewhat ad hoc manner.…”

Section: Introductionmentioning

confidence: 99%

Parameters for certain locally-regular graphs

Conder¹,

Schillewaert²,

Verret³

2021

Preprint

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“…Chapman-Linial-Peled: Families of (a, b)-regular graphs that expand both locally and globally (Polygraph constructions). [9] 2. Kaufman-Oppenheim: Bounded degree simplicial complexes arising from elementary matrix groups, which are high dimensional expanders obeying the local spectral expansion property.…”

Section: Introductionmentioning

confidence: 99%

Hyper-regular graphs and high dimensional expanders

Friedgut,

Iluz

2020

Preprint

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Let G = (V, E) be a finite graph. For d 0 > 0 we say that G is d 0 -regular, if every v ∈ V has degree d 0 . We say that G is (d 0 , d 1 )-regular, for 0for every 1 ≤ i ≤ n − 1, the joint neighborhood of every clique of size i is d i -regular); In that case, we say that G is an n-dimensional hyper-regular graph (HRG). Here we define a new kind of graph product, through which we build examples of infinite families of n-dimensional HRG such that the joint neighborhood of every clique of size at most n − 1 is connected. In particular, relying on the work of Kaufman and Oppenheim, our product yields an infinite family of n-dimensional HRG for arbitrarily large n with good expansion properties. This answers a question of Dinur regarding the existence of such objects.

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Expander Graphs — Both Local and Global

Cited by 10 publications

References 20 publications

Constructing highly regular expanders from hyperbolic Coxeter groups

Constructing highly regular expanders from hyperbolic Coxeter groups

Parameters for certain locally-regular graphs

Hyper-regular graphs and high dimensional expanders

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