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The edge-isoperimetric problem for discrete tori

Thomas A. Carlson
2002 Discrete Mathematics  
The edge-isoperimetric problem has long been solved for cartesian powers of the cycles C3 and C4, for which the lexicographic order is the optimal order, and powers of the cycles Cn with n ¿ 5, which do  ...  For powers of C5, it is clear that the lexicographic order is not optimal. We present a solution to the edge-isoperimetric problem for powers of C5 in the form of an optimal order for the vertices.  ...  The author would like to thank Sergei Bezrukov and Larry Harper for posing the problem and providing comments on the work presented here, and Dr.  ... 
doi:10.1016/s0012-365x(01)00432-0 fatcat:3ol73oh3nrb5hg5zva2innuddm

The vertex isoperimetric problem for the powers of the diamond graph

Sergei L. Bezrukov, Miquel Rius, Oriol Serra
2008 Discrete Mathematics  
We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nested solutions.  ...  We present an exact solution to the vertex isoperimetric problem on our graph by introducing a new class of orders that unifies all known isoperimetric orders defined on the cartesian powers of graphs.  ...  The compression technique we use here is a classical tool for dealing with discrete isoperimetric problems, see e.g., [8] .  ... 
doi:10.1016/j.disc.2007.04.060 fatcat:yhoairxe6bd3vge4jqjs5bqg3e

Page 4033 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
4033 05C Graph theory 2003£:05069 05C35 90C35 Carlson, Thomas A. (1-WA; Seattle, WA) The edge-isoperimetric problem for discrete tori. (English summary) Discrete Math. 254 (2002), no. 1-3, 33—49.  ...  The author considers an edge-isoperimetric problem for the Carte- sian products of cycles.  ... 

On k-ary n-cubes: theory and applications

Weizhen Mao, David M. Nicol
2003 Discrete Applied Mathematics  
In particular, the problem of characterizing the subgraph of a given number of nodes with the maximum edge count is studied.  ...  7 Many parallel processing applications have communication patterns that can be viewed as graphs called k-ary n-cubes, whose special cases include rings, hypercubes and tori.  ...  Acknowledgements 27 The authors wish to thank the referees for their helpful comments and constructive suggestions. 29  ... 
doi:10.1016/s0166-218x(02)00238-x fatcat:aeqcxwdkmfh6na2lh2ir7t4q4q

Page 5517 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
Chapter 4 (Invariant tori) concerns the problem of the existence of invariant tori and the persistence of stable tori under small perturbations of the initial system.  ...  Namely, if the isoperimetric quotient of the initial interface does not differ much from the isoperimetric quotient of the Wulff shape, then the interface shrinks to a point in finite time and the isoperi  ... 

Embedding hypercubes into cylinders, snakes and caterpillars for minimizing wirelength

Paul Manuel, M. Arockiaraj, Indra Rajasingh, Bharati Rajan
2011 Discrete Applied Mathematics  
Further, we show that the edge isoperimetric problem solves the wirelength problem of regular graphs and, in particular, hypercubes into triangular snakes and caterpillars.  ...  We consider the problem of embedding hypercubes into cylinders to minimize the wirelength.  ...  Further, this solves the wirelength problem of powers of the Petersen graph [5] and discrete tori [9] into triangular snakes and caterpillars.  ... 
doi:10.1016/j.dam.2011.07.003 fatcat:xgh2riwonng25pglaheegkomwm

Sobolev spaces on graphs

M.I. Ostrovskii
2005 Quaestiones Mathematicae. Journal of the South African Mathematical Society  
It is not difficult to show that for discrete tori the estimate from (1) is optimal, up to a multiplicative constant.  ...  11]). (2) Inequalities between discrete Sobolev seminorms (corresponding to two different edge sets and the same value of p) are very important for the problem of l p -embeddability of graph metrics, see  ... 
doi:10.2989/16073600509486144 fatcat:5eetkcnoljgazpzicgnturqo2y

Isoperimetric and Sobolev inequalities for magnetic graphs [article]

Javier Alejandro Chávez-Domínguez
2020 arXiv   pre-print
Using heat kernel techniques, we also give lower bounds for the eigenvalues of the discrete magnetic Laplacian.  ...  We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one.  ...  Acknowledgements The author thanks Profs. William B. Johnson and Keri Kornelson for helpful discussions on the subject of this paper.  ... 
arXiv:2005.10409v1 fatcat:alg2oap2m5g53mdwlqxj2zhahy

Stable periodic constant mean curvature surfaces and mesoscopic phase separation

Antonio Ros
2007 Interfaces and free boundaries (Print)  
We give a first comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry.  ...  We prove that closed, stable, constant mean curvature surfaces in R 3 /Γ , Γ ⊂ R 3 being a discrete subgroup of translations with rank k, have genus k.  ...  According to Surface Evolver experiments [22] , the isoperimetric problem in other 3-tori, like the face centered cubic torus, should admit solutions of genus 2.  ... 
doi:10.4171/ifb/168 fatcat:ubs63obb5nfjde5shk5zop5xb4

A local–global principle for vertex-isoperimetric problems

Sergei L. Bezrukov, Oriol Serra
2002 Discrete Mathematics  
We consider the vertex-isoperimetric problem (VIP) for cartesian powers of a graph G.  ...  In this paper we consider the vertex-isoperimetric problem (VIP) on graphs. Let G = (V; E) be a ÿnite simple graph, and let A ⊆ V .  ...  Acknowledgements The authors are grateful to two anonymous referees for their constructive comments that signiÿcantly improved the quality of the paper.  ... 
doi:10.1016/s0012-365x(02)00431-4 fatcat:ezit4iaclbavrnxysadhx2o6ii

Isoperimetric inequalities and Dirichlet functions of Riemann surfaces

J. M. Rodríguez
1994 Publicacions matemàtiques  
integral in the surface .  ...  We prove too, by an example, that the implication is not tru e without the condition of regularity.  ...  ~D arc cosh coth 2 The vertices of G ar e the vertices of T . The edges of G are the union of the edges of T an d the edges of G,,, , for all n~1 . The root ro of G has degree two .  ... 
doi:10.5565/publmat_38194_19 fatcat:kmferx56ijf6fiaacvpshpcpki


Irene Vecchio, Katja Schladitz, Michael Godehardt, Markus J. Heneka
2012 Image Analysis and Stereology  
However, the particle's shape is decisive for the damage it can cause, yet can not be judged reliably from 2d data.  ...  During production of mechanical components, residual dirt collects on the surfaces, thus creating a contamination that affects the durability of the assembled products.  ...  Note that the discretization chosen for the cylinders plays a big role in all the feature estimations, too.  ... 
doi:10.5566/ias.v31.p163-174 fatcat:x7uslhjkvfa2xn6ddjjteowr5u

The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces [article]

Brian Benson
2016 arXiv   pre-print
To do this, we use results of Hass-Morgan for the isoperimetric problem of these manifolds. We also give an example of a finite area 2-manifold where no such subset exists.  ...  We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian n-manifolds.  ...  The author would also like to thank his advisor Nathan Dunfield for suggesting this project as well as for many helpful suggestions and insights.  ... 
arXiv:1509.08993v2 fatcat:xgvpvn3cnrhxpfe2ry64jlqgcu

On spanning tree congestion of graphs

Kyohei Kozawa, Yota Otachi, Koichi Yamazaki
2009 Discrete Mathematics  
In this paper, we show the spanning tree congestion for the complete k-partite graphs and the two-dimensional tori.  ...  For e ∈ E(T ), the congestion of e is the number of edges in G connecting two components of T − e. The edge congestion of G in T is the maximum congestion over all edges in T .  ...  In Section 4, we show the spanning tree congestion for the two-dimensional tori. This problem is related to Hruska's result for the two-dimensional grids [12] .  ... 
doi:10.1016/j.disc.2008.12.021 fatcat:rstajzdu7bh7rldhe7ruaxuaqu

Page 4844 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
Summary: “We study the problem of partitioning point sets in the space so that each equivalence class is a convex polytope disjoint from the others. For a set of m points P in R?  ...  The lower bound also holds for partitions into empty convex CONVEX AND DISCRETE GEOMETRY polytopes.”  ... 
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