## Discrete isoperimetric inequalities (2004)

Venue: | Surveys in Differential Geometry IX, International Press, 53–82 |

Citations: | 10 - 1 self |

### BibTeX

@INPROCEEDINGS{Chung04discreteisoperimetric,

author = {Fan Chung},

title = {Discrete isoperimetric inequalities},

booktitle = {Surveys in Differential Geometry IX, International Press, 53–82},

year = {2004},

pages = {53--82},

publisher = {Press}

}

### OpenURL

### Abstract

### Citations

1373 | Introduction to Parallel Algorithms and Architectures - Leighton - 1992 |

1221 | Graph Theory - Bollobás - 1979 |

418 |
The theory of matrices
- Lancaster, Tismenetsky
(Show Context)
Citation Context ... ′ denote the submatrix obtained by deleting the vth row and vth column of L. Since L = BB∗ ,wehave L ′ =B0B∗ 0 where B0 denotes the submatrix of B without the vth column. By the Binet-Cauchy Theorem =-=[53]-=- we have det B0B ∗ 0 = � X C0 det BX det B ∗ X where the sum ranges over all possible choices of size n − 1 subsets X of E(G) andBXdenotes the square submatrix of B0 whose n − 1 columns correspond to ... |

401 | A separator theorem for planar graphs - Lipton, Tarjan - 1979 |

349 | A random graph model for massive graphs
- Aiello, Chung, et al.
- 2000
(Show Context)
Citation Context ...associated with stability of chemicals [9]. In recent years, many realistic graphs that arise in Internet and biological networks can be modeled as graphs with certain “power law” degree distribution =-=[1, 2, 3]-=-. Again, eigenvalues come into play since random graphs with given expected degrees are shown to have eigenvalue distribution as predicted [32, 33]. In this paper, we discuss only a few applications o... |

328 | Eigenvalues and Expanders
- Alon
- 1986
(Show Context)
Citation Context ..., d, λ)-graphs (i.e., regular graphs on n vertices having degree d with all but one eigenvalue of the adjacency matrix bounded above by λ). Such graphs are extensively examined in many papers by Alon =-=[4]-=- and others [46, 60]. There is a recent comprehensive survey by Krivelevich and Sudakov [46] on (n, d, λ)-graphs. Here we deal with general graphs with no degree constraints. Throughout the paper, we ... |

306 |
Group Representations in Probability and Statistics
- Diaconis
- 1988
(Show Context)
Citation Context ...rlier approaches on discrete isoperimetric inequalities focuses on discretizations of manifolds [35, 42]. Another approach is to study graphs with group symmetry [58] or random walks on finite groups =-=[36]-=-. In this paper, we consider general graphs and our approach here is from a graph-theoretic point of view. 2s2 Combinatorial and normalized Laplacian One of the classical results in graph theory is th... |

299 | Approximating the permanent - Jerrum, Sinclair - 1989 |

241 | An Approximate Max–Flow Min–Cut Theorem for Uniform Multicommodity Flow Problems with Application to Approximation Algorithm - Leighton, Rao - 1988 |

176 | λ1, isoperimetric inequalities for graphs, and superconcentrators - Alon, Milman - 1985 |

115 | Logarithmic Sobolev inequalities for finite Markov chains
- Diaconis, Saloff-Coste
- 1996
(Show Context)
Citation Context ...for general graphs. Logarithmic Sobolev inequalities The upper bound for the rate of convergence in Theorem 19 can sometimes be further improved by using the log-Sobolev constant α defined as follows =-=[37]-=-: � x∼y α(G) = α = inf f�=0 (f(x) − f(y))2wx,y � x f 2 (x)dx log f 2 (x)vol(G) where f ranges over all nontrivial vectors f : V → R. Then we have the following [18]: È z f 2 (z)dz Theorem 20 For a wei... |

114 |
Über die Auflösung der Gleichungen, auf welche man bei der untersuchung der linearen Verteilung galvanischer ströme geführt wird
- Kirchhoff
(Show Context)
Citation Context ...l graphs and our approach here is from a graph-theoretic point of view. 2s2 Combinatorial and normalized Laplacian One of the classical results in graph theory is the matrix-tree theorem by Kirchhoff =-=[43]-=-, which states that the number of spanning trees in a graph is determined by the determinant of a principle minor of the combinatorial Laplacian. For a graph G with vertex set V and edge set E, the co... |

107 | The spectra of random graphs with given expected degrees. Internet Mathematics 1(3):257–275
- Chung, L, et al.
- 2004
(Show Context)
Citation Context ...graphs with certain “power law” degree distribution [1, 2, 3]. Again, eigenvalues come into play since random graphs with given expected degrees are shown to have eigenvalue distribution as predicted =-=[32, 33]-=-. In this paper, we discuss only a few applications of isoperimetric inequalities. It would be of interest to find further applications especially for power law graphs. Acknowledgement: The author wis... |

90 | Quasi-random graphs, Combinatorica 9 - Chung, Graham, et al. - 1989 |

81 | Isoperimetric number of graphs - Mohar - 1989 |

74 |
Hamiltonian circuits in random graphs
- Pósa
- 1976
(Show Context)
Citation Context ...t least ɛd. The applications on random walks in Section 14 will use the above eigenvalue inequality. 10 Paths and cycles One of the major theorems in studying the paths of a graph is a result of Pósa =-=[57]-=- (see [54], Problem 10.20, for an elegant solution). Pósa’s Theorem In a graph H if every subset X of vertices with |X| ≤ksatisfies then H contains a path with 3k − 2 vertices. |δ(X)| ≥2|X|−1, Theorem... |

56 |
Graph Theory 1736-1936
- Biggs, Lloyd, et al.
- 1976
(Show Context)
Citation Context ...h theory has 250 years of history. In the very early days, graphs were used to study the structure of molecules and in particular, the eigenvalues of graphs are associated with stability of chemicals =-=[9]-=-. In recent years, many realistic graphs that arise in Internet and biological networks can be modeled as graphs with certain “power law” degree distribution [1, 2, 3]. Again, eigenvalues come into pl... |

54 |
Expanding graphs contain all small trees, Combinatorica 7(1
- Friedman, Pippenger
- 1987
(Show Context)
Citation Context ...rsal graphs for trees of bounded degrees There is quite a literature on so-called “universal graphs” that contain all trees on n vertices or other families of graphs such as trees with bounded degree =-=[5, 8, 16, 20, 39]-=-. One of the main avenues in the study of universal graphs is the connection with expanding properties of the graph. Friedman and Pippenger [39] proved the following beautiful result: Theorem [39] Sup... |

47 | Eigenvalues of random power law graphs
- Chung, Lu, et al.
(Show Context)
Citation Context ...graphs with certain “power law” degree distribution [1, 2, 3]. Again, eigenvalues come into play since random graphs with given expected degrees are shown to have eigenvalue distribution as predicted =-=[32, 33]-=-. In this paper, we discuss only a few applications of isoperimetric inequalities. It would be of interest to find further applications especially for power law graphs. Acknowledgement: The author wis... |

42 | Diameters and Eigenvalues - Chung - 1989 |

39 |
isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds
- Kanai, Rough
- 1985
(Show Context)
Citation Context ...Riemannian geometry have been long studied and well developed (see [12, 64]). As a result, one of the earlier approaches on discrete isoperimetric inequalities focuses on discretizations of manifolds =-=[35, 42]-=-. Another approach is to study graphs with group symmetry [58] or random walks on finite groups [36]. In this paper, we consider general graphs and our approach here is from a graph-theoretic point of... |

36 | R.L.Graham. Routing permutations on graphs via matching - Alon, Chung - 1994 |

35 | Quasi-random set systems - Chung, Graham - 1991 |

31 |
Variétés riemanniennes isométriques à l’infini
- Coulhon, Saloff-Coste
- 1995
(Show Context)
Citation Context ...Riemannian geometry have been long studied and well developed (see [12, 64]). As a result, one of the earlier approaches on discrete isoperimetric inequalities focuses on discretizations of manifolds =-=[35, 42]-=-. Another approach is to study graphs with group symmetry [58] or random walks on finite groups [36]. In this paper, we consider general graphs and our approach here is from a graph-theoretic point of... |

29 | Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs
- Chung, Grigor’yan, et al.
(Show Context)
Citation Context ...igenvalue bounds above are analogs of the Polyá conjecture for Dirichlet eigenvalues of a regular domain M. λk ≥ 2π k 2 ( ) n vol M wn where wn is the volume of the unit disc in Rn . In a later paper =-=[23]-=-, the condition in (7) is further relaxed. It was shown that if in a graph G =(V,E), any subset X ⊆ V satisfies e(X, ¯ X) ≥ c(vol(X)) (δ−1)/δ for vol(X) ≤ c1, then the Dirichlet eigenvalue λk(S) for t... |

27 |
Isoperimetric inequalities: differential geometric and analytic perspectives”, Cambridge Tracts
- Chavel
- 2001
(Show Context)
Citation Context ...discrete isoperimetric inequalities and their continuous counterpart, as evidenced in Section 7 to 9. Isoperimetric inequalities for Riemannian geometry have been long studied and well developed (see =-=[12, 64]-=-). As a result, one of the earlier approaches on discrete isoperimetric inequalities focuses on discretizations of manifolds [35, 42]. Another approach is to study graphs with group symmetry [58] or r... |

26 |
On size Ramsey number of paths, trees, and circuits
- Beck
- 1983
(Show Context)
Citation Context ...rsal graphs for trees of bounded degrees There is quite a literature on so-called “universal graphs” that contain all trees on n vertices or other families of graphs such as trees with bounded degree =-=[5, 8, 16, 20, 39]-=-. One of the main avenues in the study of universal graphs is the connection with expanding properties of the graph. Friedman and Pippenger [39] proved the following beautiful result: Theorem [39] Sup... |

26 | An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian
- Chung, Faber, et al.
- 1994
(Show Context)
Citation Context .... Here, λ is λ = λ1 if 1 − λ1 ≥ λn−1 − 1. � � log(n − 1) D(G) ≤ . log(1/(1 − λ)) (5) 6.1 Eigenvalues and diameters Inequality (5) can be generalized to all graphs by using the combinatorial Laplacian =-=[19]-=-. Theorem 9 Suppose a graph G on n vertices has eigenvalues 0 ≤ σ1 ≤ ... ≤ σn−1. The diameter of G satisfies ⎡ ⎢ log(n − 1) D(G) ≤ ⎢ ⎢log σn−1 ⎤ ⎥ + ⎥ . σ1 ⎥ σn−1 − σ1 We note that for some graphs the... |

25 | Sparse pseudo-random graphs are Hamiltonian
- Krivelevich, Sudakov
(Show Context)
Citation Context ...1 1 (1 − θ2k−2 n (d2 θ2 )t−1 ). 2(2k − 1)θn d ≥ (1 + o(1)) d2k−1 1 (1 − θ2k−2 n (d2 θ2 )t−1 ). d n ≥ (1 + o(1)) 2k 2(2k − 1)θ2k−1 which is a contradiction to the assumption that n ≪ d2k /θ2k−1 . � In =-=[45]-=- Krivelevich and Sudakov showed that a d-regualar graph on n vertices is Hamiltonian if the eigenvalues of the combinatorial Laplacian satisfy (log log n) |d − σi| ≤c 2 log n(log log log n) d for i �=... |

19 |
Random evolution in massive graphs, Handbook of Massive Data Sets, Volume 2, (Eds
- Aiello, Chung, et al.
- 2001
(Show Context)
Citation Context ...associated with stability of chemicals [9]. In recent years, many realistic graphs that arise in Internet and biological networks can be modeled as graphs with certain “power law” degree distribution =-=[1, 2, 3]-=-. Again, eigenvalues come into play since random graphs with given expected degrees are shown to have eigenvalue distribution as predicted [32, 33]. In this paper, we discuss only a few applications o... |

18 | A lower bound for the smallest eigenvalue - Cheeger - 1970 |

17 | Upper bounds for eigenvalues of the discrete and continuous Laplace operators
- Chung, Grigor’yan, et al.
- 1996
(Show Context)
Citation Context ...unds for manifolds ⎤ ⎥ � vol ¯ Xi vol ¯ Xj volXi volXj log λn−j−1+λk−j λn−j−1−λk−j The above discrete methods can be used to derive new eigenvalue upper bounds for compact smooth Riemannian manifolds =-=[21, 22]-=-. Let M be a complete Riemannian manifold with finite volume and let L be the self-adjoint operator −∆, where ∆ is the Laplace operator associated with the Riemannian metric on M. Or, we could conside... |

17 |
A geometric approach to on-diagonal heat kernel lower bounds on groups
- Coulhon, Grigor’yan, et al.
(Show Context)
Citation Context ...values and the heat kernel techniques to obtain eigenvalue lower bounds. In the literature, there are many general formulations for discrete heat kernel in connection with the continuous heat kernels =-=[34, 40]-=-. Here we define a natural heat kernel for general graphs. Let φi denote the eigenfunction for the Laplacian corresponding to eigenvalue λi. We now define the heat kernel of S as a n × n matrix Ht = �... |

15 | On universal graphs for spanning trees
- Chung, Graham
- 1983
(Show Context)
Citation Context ...rsal graphs for trees of bounded degrees There is quite a literature on so-called “universal graphs” that contain all trees on n vertices or other families of graphs such as trees with bounded degree =-=[5, 8, 16, 20, 39]-=-. One of the main avenues in the study of universal graphs is the connection with expanding properties of the graph. Friedman and Pippenger [39] proved the following beautiful result: Theorem [39] Sup... |

13 | On cycle-complete graph Ramsey numbers
- Erdős, Faudree, et al.
- 1978
(Show Context)
Citation Context ...of the combinatorial Laplacian satisfy |d − σi| ≤θfor i �= 0. If d 2k ≫ nθ 2k−1 ,thenGcontains a cycle of length 2k +1,ifnis sufficiently large. Proof: We consider δi(v) ={u : d(u, v) =i}. In a paper =-=[38]-=- by Erdős et al., it was shown that if a graph G contains no cycle of length 2k + 1, then for any 1 ≤ i ≤ k the induced subgraph on δi(v) contains an independent set S of size at least |δi(v)|/(2k − 1... |

12 |
On sampling with Markov chains. Random Structures &Algorithms
- Chung, Graham, et al.
- 1996
(Show Context)
Citation Context ...wo subsets X, Y of vertices in G, the distance between X and Y , denoted by d(X, Y ), is the minimum distance between a vertex in X and a vertex in Y ; i.e., d(X, Y ) = min{d(x, y) : x∈X, y ∈ Y }. In =-=[31]-=-, the distance between two sets can be related to eigenvalues as follows: Theorem 10 Suppose G is not a complete graph. For X, Y ⊂ V (G), ⎡ � ⎤ volXvol ¯ Y¯ log d(X, Y ) ≤ ⎢ volX volY ⎥ ⎢ ⎥ . (6) ⎢ ⎥ ... |

11 | S.T.,On the parabolic kernel of the Schrödinger operator, ActaMath - Li - 1986 |

9 |
Eigenvalue inequalities for graphs and convex subgraphs
- Chung, Yau
- 1997
(Show Context)
Citation Context ...d that 1 − ρ ≥ λ1(S), where λ1(S) denotes the first Neumann eigenvalue of the induced subgraph S of Γ. In particular, if the total row sum (minus the maximum row sum) is ≥ c ′ n 2 ,itcanbe shown (see =-=[27]-=-) that λ1(S) ≥ c kD 2 ,whereDis the diameter of S. This implies that a random walk converges to the uniform distribution in O(k 2 D 2 (log n)) steps. (In some cases, the factor log n can be further re... |

9 | Arithmetic groups and graphs without short cycles, 6th Intern - Margulis - 1984 |

8 |
Universal graphs and induced-universal graphs
- Chung
- 1990
(Show Context)
Citation Context |

8 | A Harnack inequality for homogeneous graphs and subgraphs
- Chung, Yau
- 1994
(Show Context)
Citation Context ...njugation by elements of K, i.e., for all a ∈ K, aKa−1 = K. Let f denote an eigenfunction in an invariant homogeneous graph with edge-generating set K consisting of k generators. Then it can be shown =-=[26]-=- that 1 k � (f(x) − f(ax)) 2 ≤ 8λ sup f 2 (y). a∈K An induced subgraph S of a graph Γ is said to be strongly convex if for all pairs of vertices u and v in S, all shortest paths joining u and v in Γ a... |

6 |
Laplacians of graphs and Cheeger’s inequalities. Combinatorics, Paul Erdös is Eighty
- Chung
- 1996
(Show Context)
Citation Context ...¯ X). where ¯ X is the complement of X. 11s5.1 Isoperimetric inequalities for edge boundaries The edge boundary is closely related to the discrete Cheeger’s constant, which is defined as follows (see =-=[15, 13]-=-). |∂(X)| hG = inf X min{vol(X), vol( ¯ X)} . The eigenvalues of the normalized Laplacian are related to Cheeger’s constant by the discrete Cheeger inequality: 2hG ≥ λ1 ≥ h2G 2 . Clearly, this implies... |

6 |
Heat kernels on manifolds, graphs and fractals
- Grigor’yan
- 2003
(Show Context)
Citation Context ...values and the heat kernel techniques to obtain eigenvalue lower bounds. In the literature, there are many general formulations for discrete heat kernel in connection with the continuous heat kernels =-=[34, 40]-=-. Here we define a natural heat kernel for general graphs. Let φi denote the eigenfunction for the Laplacian corresponding to eigenvalue λi. We now define the heat kernel of S as a n × n matrix Ht = �... |

6 |
Pseudo-random graphs. preprint
- Krivelevich, Sudakov
(Show Context)
Citation Context ...(i.e., regular graphs on n vertices having degree d with all but one eigenvalue of the adjacency matrix bounded above by λ). Such graphs are extensively examined in many papers by Alon [4] and others =-=[46, 60]-=-. There is a recent comprehensive survey by Krivelevich and Sudakov [46] on (n, d, λ)-graphs. Here we deal with general graphs with no degree constraints. Throughout the paper, we consider only finite... |

5 |
Explicit constructions of linear-sized tolerant networks
- Alon, Chung
- 1988
(Show Context)
Citation Context |

4 | Eigenvalues and diameters for manifolds and graphs
- Chung, Grigor’yan, et al.
(Show Context)
Citation Context ...unds for manifolds ⎤ ⎥ � vol ¯ Xi vol ¯ Xj volXi volXj log λn−j−1+λk−j λn−j−1−λk−j The above discrete methods can be used to derive new eigenvalue upper bounds for compact smooth Riemannian manifolds =-=[21, 22]-=-. Let M be a complete Riemannian manifold with finite volume and let L be the self-adjoint operator −∆, where ∆ is the Laplace operator associated with the Riemannian metric on M. Or, we could conside... |

3 |
Logarithmic Sobolev techniques for random walks on graphs
- Chung
- 1996
(Show Context)
Citation Context ...bolev constant α defined as follows [37]: � x∼y α(G) = α = inf f�=0 (f(x) − f(y))2wx,y � x f 2 (x)dx log f 2 (x)vol(G) where f ranges over all nontrivial vectors f : V → R. Then we have the following =-=[18]-=-: È z f 2 (z)dz Theorem 20 For a weighted graph G with log-Sobolev constant α, there is a lazy walk satisfying provided that where λ is as defined in Theorem 19. ∆(t) <e 2−c t ≥ 1 vol(G) log log + α m... |

2 |
Chung and Prasad Tetali, Isoperimetric inequalities for cartesian products of graphs, 7
- K
- 1998
(Show Context)
Citation Context ...ons f : V → R which are not identically zero. x∈V 12 (3)sA variation of (3) seems to be particularly useful, e.g., for deriving isoperimetric relationships between graphs and their Cartesian products =-=[24]-=-. hG ≥ inf f�=0 where f : V (G) → R satisfies � f(x)dx =0. x∈V � |f(x) − f(y)| x∼y � |f(x)|dx x∈V ≥ 1 2 hG 5.2 Isoperimetric inequalities for vertex boundaries We will prove the following basic isoper... |

1 |
A spectral Turán theorem, preprint
- Chung
(Show Context)
Citation Context ...e d is t-Turán if the second largest eigenvalue of its adjacency matrix λ is sufficiently small. Their result can be extended to general graphs by using the isoperimetric and discrepancy inequalities =-=[17]-=-. Theorem 17 Suppose a graph G on n vertices has eigenvalues of the normalized Laplacian 0=λ0≤λ1≤...≤λn−1 with ¯ λ =maxi�=0 |1 − λi| satisfying ¯λ 1 = o( ) (11) vol−2t+1(G)vol(G) t−1 where voli(S) = �... |

1 |
amd E. Szemerédi, Limit distributions for the existence of Hamilton circuits in a random graph
- Komlós
- 1983
(Show Context)
Citation Context ...inatorial Laplacian satisfy (log log n) |d − σi| ≤c 2 log n(log log log n) d for i �= 0 and for some constant c. The method is a modified version of Posa’s technique developed by Komlós and Szemerédi =-=[44]-=- for examining Hamiltonian cycles in random graphs. By using the discrepancy inequalities and the isoperimetric inequalities in previous sections, the above result can be extended to general graphs as... |