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Isoperimetric upper bound for the first eigenvalue of discrete Steklov problems
[article]

2020
*
arXiv
*
pre-print

For finite

arXiv:2002.08751v2
fatcat:yehw3p5kenhsvch643sxywqbfe
*graphs*with boundary included*in*a*Cayley**graph*associated to a group of polynomial growth, we give*an*upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices ... This extends recent results of Han and Hua, who obtained a similar result*in*the case of ℤ^n. We obtain the result using metric properties of*Cayley**graphs*associated to groups of polynomial growth. ...*Isoperimetric*upper bound for σ 1*in**Cayley**graphs*with polynomial growth*In*this section, we prove the results presented*in*the introduction and give examples of application. ...##
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On the isoperimetric number of a k-degree Cayley graph

2009
*
Applied Mathematics Letters
*

*In*this work, we shall concentrate on the

*isoperimetric*properties of the k-degree

*Cayley*

*graphs*G k,n , which were proposed recently for building interconnection networks. ... We shall give the exact

*isoperimetric*number i(G k,n ) when n = 2, and

*an*upper bound of i(G k,n )

*in*the general case. ... Additionally, we give

*an*upper bound of the

*isoperimetric*number of the k-degree

*Cayley*

*graph*, from structure analysis.

*In*the final section, we indicate future work on this

*problem*. ...

##
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Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

2020
*
Journal of Geometric Analysis
*

For finite

doi:10.1007/s12220-020-00572-2
fatcat:xxw5r57iqbhwtkdztzlodldqsu
*graphs*with boundary included*in*a*Cayley**graph*associated to a group of polynomial growth, we give*an*upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices ... AbstractWe study upper bounds for the first non-zero eigenvalue of the Steklov*problem*defined on finite*graphs*with boundary. ...*Isoperimetric*Upper Bound for 1*in**Cayley**Graphs*with Polynomial Growth*In*this section, we prove the results presented*in*the introduction and give examples of application. ...##
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Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality
[article]

2003
*
arXiv
*
pre-print

Let G be a

arXiv:math/0303127v1
fatcat:jrz5lflphrb7zo2skkpbci5b6y
*graph*which satisfies c^-1 a^r < |B(v,r)| < c a^r, for some constants c,a>1, every vertex v and every radius r. ... We prove that this implies the*isoperimetric*inequality |∂ A| > C |A| / (2+ |A|) for some constant C=C(a,c) and every finite set of vertices A. ... Coulhon and Saloff-Coste [4] proved that when G is a*Cayley**graph*of*an*infinite, finitely-generated, group, the*isoperimetric*inequality ∂A ≥ |A| 4 m φ 2|A| (2) holds for every finite A ⊂ V (G), where ...##
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Expander families, group structure, and semidirect products

2013
*
Irish Mathematical Society Bulletin
*

*Cayley*

*graphs*. ... It is

*an*interesting and open question to determine which groups yield

*Cayley*

*graphs*that form expander families. ... It is

*an*open

*problem*to find necessary and sufficient conditions for a sequence of finite groups to admit

*an*expander family as a sequence of

*Cayley*

*graphs*. The class of 22 M. AIVAZIAN AND M. ...

##
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Evacuation schemes on Cayley graphs and non-amenability of groups
[article]

2021
*
arXiv
*
pre-print

*In*this paper we introduce a concept of

*an*evacuation scheme on the

*Cayley*

*graph*of

*an*infinite finitely generated group. ... We obtain a criterion for existing of such a scheme

*in*terms of

*isoperimetric*constant of the

*graph*. We analyze R. Thompson's group F, for which the amenability property is a famous open

*problem*. ...

*In*this case we say that the family (p v ) v∈G is

*an*evacuation scheme on the

*Cayley*

*graph*Γ. ...

##
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Isoperimetry in integer lattices

2018
*
Discrete Analysis
*

We solve the edge

doi:10.19086/da.3555
fatcat:dcnr2avcgnbafk4nblcxs2pm3i
*isoperimetric**problem*asymptotically for every*Cayley**graph*on Z^d. ... The edge*isoperimetric**problem*for a*graph*G is to determine, for each n, the minimum number of edges leaving any set of n vertices. ... Relation to continuous*problems**In*this section we indicate connections between*isoperimetric**problems**in**Cayley**graphs*on Z d and classical results from convex geometry. ...##
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The firefighter problem on polynomial and intermediate growth groups
[article]

2020
*
arXiv
*
pre-print

We prove that any

arXiv:2002.11205v1
fatcat:y6tgccx6hrdklgo54a55fpljo4
*Cayley**graph*G with degree d polynomial growth does not satisfy {f(n)}-containment for any f=o(n^d-2). ... This settles the asymptotic behaviour of the firefighter*problem*on such*graphs*as it was known that Cn^d-2 firefighters are enough, answering and strengthening a conjecture of Develin and Hartke. ... We will identify G with its*Cayley**graph*w.r.t. S. ...##
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Uniform non-amenability, cost, and the first ℓ2-Betti number

2008
*
Groups, Geometry, and Dynamics
*

*An*ergodic version h e ./ of the uniform

*isoperimetric*constant h./ is defined as the infimum over all essentially free ergodic and measure preserving actions˛of of the uniform

*isoperimetric*constant h.R ... We then define

*isoperimetric*constants

*in*the framework of measured equivalence relations. ... It takes values

*in*OE0; 1 and is defined as

*an*infimal value of

*isoperimetric*ratios for 'finite sets'

*in*the

*Cayley*

*graphs*of R (see Section 4) . ...

##
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Uniform non-amenability, cost, and the first l^2-Betti number
[article]

2008
*
arXiv
*
pre-print

*An*ergodic version h_e() of the uniform

*isoperimetric*constant h() is defined as the infimum over all essentially free ergodic and measure preserving actions α of of the uniform

*isoperimetric*constant ... For

*an*ergodic measured equivalence relation R of type , the uniform

*isoperimetric*constant h(R) of R is invariant under orbit equivalence and satisfies 2β_1(R)≤ 2C(R)-2≤ h(R), where β_1() is the first ... The second author was supported by

*an*EPDI Post-doctoral Fellowship and is grateful to IHES for its hospitality. ...

##
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Book Review: Isoperimetric inequalities: Differential geometric and analytic perspectives

2002
*
Bulletin of the American Mathematical Society
*

A more contemporary direction would be to forget about manifolds entirely, or at least put them

doi:10.1090/s0273-0979-02-00954-0
fatcat:stvgj2hijfgn5ogqtxa5gmbzia
*in*the background, and look at*graphs*, particularly*Cayley**graphs*of groups. ... The resulting surface looks like the*graph*if you stand far enough away and blur your vision. This suggests that defining and understanding*isoperimetric**problems*on*graphs*might be a good idea. ...##
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Page 3571 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

*In*this article the author proves

*an*anal- ogous result for abelian groups of even order. ... Much of this interest has been focused on the

*Cayley*isomorphism

*problem*, which asks for necessary and sufficient conditions for two

*Cayley*

*graphs*on the same group to be isomorphic. ...

##
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Functions on groups and computational complexity
[article]

2002
*
arXiv
*
pre-print

those functions and the computational complexity of the word

arXiv:math/0202124v1
fatcat:euxlke4i2bcavfpvgmbhxb4bee
*problem*(deterministic time, nondeterministic time, symmetric space). ... We give some connections between various functions defined on finitely presented groups (*isoperimetric*, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between ... Moreover, the partial*Cayley**graph*TC n coincides with the*Cayley**graph*within radius n/2. 2 Another way to build a 2-complex*in*order to solve the word*problem*for words of length ≤ n is as follows: For ...##
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Page 1678 of Mathematical Reviews Vol. , Issue 2000c
[page]

2000
*
Mathematical Reviews
*

*An*embedding of a

*Cayley*digraph D is said to be stiff if there is

*an*arc rotation e such that the arc rotation at any vertex of D is either c or e~! ... Our method gives partial answers to the open

*problems*| and 2

*in*[S. Hong, J. H. Kwak and J. ...

##
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Page 6815 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

and extremal

*problems*for directed and oriented*graphs*. ... If*in*addition S =S~', that is, s € S implies s~' € S, then the*Cayley**graph*X(G;S) is defined to be the*graph*having vertex set G with*an*edge between g and A if and only if hg~' € S. ...
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