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

Hélène Perrin
2020 arXiv   pre-print
For finite 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.  ... 
arXiv:2002.08751v2 fatcat:yehw3p5kenhsvch643sxywqbfe

On the isoperimetric number of a k-degree Cayley graph

Zhantao Huang, Yinglie Jin, Ke Liang
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.  ... 
doi:10.1016/j.aml.2009.01.056 fatcat:n4f6467zvrdfxfu3c3tw4vqvoq

Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

Hélène Perrin
2020 Journal of Geometric Analysis  
For finite 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.  ... 
doi:10.1007/s12220-020-00572-2 fatcat:xxw5r57iqbhwtkdztzlodldqsu

Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality [article]

Itai Benjamini, Oded Schramm
2003 arXiv   pre-print
Let G be a 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  ... 
arXiv:math/0303127v1 fatcat:jrz5lflphrb7zo2skkpbci5b6y

Expander families, group structure, and semidirect products

Matthew Aivazian, Mike Krebs
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.  ... 
doi:10.33232/bims.0071.21.30 fatcat:jaoqkvak3fhvjdpxnudvkpqqxu

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

Victor Guba
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.  ... 
arXiv:2112.09812v1 fatcat:vfg26sennfcitbu5mr6qdjv4t4

Isoperimetry in integer lattices

Ben Barber, Joshua Erde
2018 Discrete Analysis  
We solve the edge 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.  ... 
doi:10.19086/da.3555 fatcat:dcnr2avcgnbafk4nblcxs2pm3i

The firefighter problem on polynomial and intermediate growth groups [article]

Gideon Amir, Rangel Baldasso, Gady Kozma
2020 arXiv   pre-print
We prove that any 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.  ... 
arXiv:2002.11205v1 fatcat:y6tgccx6hrdklgo54a55fpljo4

Uniform non-amenability, cost, and the first ℓ2-Betti number

Russell Lyons, Mikaël Pichot, Stéphane Vassout
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) .  ... 
doi:10.4171/ggd/49 fatcat:qdi4ffgtofay7cfcsq5gdzs5ty

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

Russell Lyons, Mikaël Pichot, Stéphane Vassout
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.  ... 
arXiv:0711.0393v2 fatcat:jpaaboy7ifex3jmojed6op6osq

Book Review: Isoperimetric inequalities: Differential geometric and analytic perspectives

Robert Brooks
2002 Bulletin of the American Mathematical Society  
A more contemporary direction would be to forget about manifolds entirely, or at least put them 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.  ... 
doi:10.1090/s0273-0979-02-00954-0 fatcat:stvgj2hijfgn5ogqtxa5gmbzia

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.  ... 

Functions on groups and computational complexity [article]

Jean-Camille Birget
2002 arXiv   pre-print
those functions and the computational complexity of the word 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  ... 
arXiv:math/0202124v1 fatcat:euxlke4i2bcavfpvgmbhxb4bee

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.  ... 

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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