N. Alon and V. D. Milman. 1 , isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38:73--88, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is c j : Recall also that Delta denotes the maximum degree of a vertex, and that given a set of vertices S, we denote by ffi(S) the set of edges with one endpoint in S and the other in V n S. We make use of the following elementary result, where the lower bound is due to Alon and Milman [1] and the upper bound is due to Juvan and Mohar [22] Lemma 3.1. Let S ae V be a subset of the vertices of a graph G. Then jffi(S)j n (Q) n : 2 Theorem 3.2. The envelope size of a symmetric matrix A can be bounded in terms of the eigenvalues of the associated Laplacian matrix as ....

N. Alon and V. Milman, 1 , isoperimetric inequalities for graphs, and superconcentrators, J. of Combin. Theory, Series B, 38 (1985), pp. 73 -- 88.

....G. The associated eigenvalue # 2 is the algebraic connectivity of G, which is closely related to the vertex and edge connectivities of G: # 2 v(G) e(G) 2) A general method for obtaining asymptotic isoperimetric inequalities for families of graphs based on # 2 is developed is described in [1]. A Cheeger like inequality has been shown in, e.g. 19] As for manifolds, the nodal domain theorem for graphs does not provide a sharp inequality for all graphs. For manifolds equality for every eigenvalue holds only in dimension one, i.e. for a string. For spheres with the standard metric a ....

N. Alon and V. D. Milman. # 1 , isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory, Ser. B, 38:73--88, 1985.

....to include recursion. It is worth noting that spectral methods have not been limited to graph partitioning; work has been done using the spectrum of the adjacency matrix in graph coloring [4] and using the Laplacian spectrum to prove theorems about expander graph and superconcentrator properties [3] [1] 2] The work on expanders has explored the relationship of # 2 to the isoperimetric number; Mohar has given an upper bound on the isoperimetric number using a strong discrete version of the Cheeger inequality [22] Reference [8] is a book length treatment of graph spectra, and it predates ....

N. Alon and V. D. Milman, # 1 , isoperimetric inequalities for graphs, and superconcentrators, Journal of Combinatorial Theory, Series B, 38 (1985), pp. 73--88.

.... both because # 2 occurs in bounds on cut quality [24] and because they can be used in isolating structural properties of the eigenvectors used in making the cuts [16, 28] The eigenvalue # 2 has been related to expansion properties of graphs and can be used in determining if a graph is an expander [1, 2]. One common class of techniques for computing such lower bounds uses properties of graph embeddings [9, 15, 20, 22, 26, 27] In such methods, a graph H is embedded into the graph G under study; that is, vertices of H are identified with vertices of G, and paths in G are specified to correspond ....

N. Alon and V. D. Milman, # 1 , isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B, 38 (1985), pp. 73--88.

....Janina Werner [24] 2 Expansion and Eigenvalues The aim of the present section is to explain the connection between the edge expansion of a graph and the second largest eigenvalue of a certain matrix, which will be relevant in Section 3. This connection originates in Alon s and Milman s work [4, 5] and was speci cally adapted for our context by Aldous [3] Our treatment closely follows Behrend s book [7] Let G = V; E) be a graph (without loops or multiple edges) on n : jV j nodes. We de ne a random walk (i.e. transition probabilities for all edges in both directions) on G in a ....

N. Alon and V. D. Milman.  1 , isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory, Ser. B, 38:73-88, 1985.

.... 2 occurs in bounds on cut quality [Moh89] and because they can be used in isolating structural properties of the eigenvectors used in making the cuts [GM95, ST96] The eigenvalue 2 has been related to expansion properties of graphs, and can be used in determining if a graph is an expander [AM85a, Alo86]. In many of the preceding applications, it is necessary to show bounds for classes of graphs in order to state results in terms of asymptotic algorithm behavior. For example, in the analysis of spectral partitioning, GM95] uses eigenvalue bounds on a family of boundeddegree graphs to prove ....

N. Alon and V. D. Milman. 1 , isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38:73--88, 1985.

....NSF Grant CCR 9505472, DARPA contract N00014 95 1246, and Army Contract DAAH04 95 1 0607. 1 structural properties of the eigenvectors used in making the cuts [16, 28] The eigenvalue # 2 has been related to expansion properties of graphs, and can be used in determining if a graph is an expander [1, 2]. One common class of techniques for computing such lower bounds uses properties of graph embeddings [9, 15, 20, 22, 26, 27] In such methods, a graph H is embedded into the graph G under study; that is, vertices of H are identified with vertices of G, and paths in G are specified to correspond to ....

N. Alon and V. D. Milman, # 1 , isoperimetric inequalities for graphs, and superconcentrators, Journal of Combinatorial Theory, Series B, 38 (1985), pp. 73--88.

.... expanders arises as a family of Cayley graphs of the simple groups PSL(2; q) Margulis [Ma] Lubotzky Phillips Sarnak [LPS] Some other families of nite simple groups have also been known to give rise to families of expanders (notably, PSL(d; q) for xed dimension d 3, see Alon Milman [AM]) It is not known, however, whether or not all nite simple groups admit bounded degree expander Cayley graphs. Even the seemingly most accessible cases are open. Problem 2.2. Do the alternating groups and the linear groups PSL(d; q) for xed q admit families of bounded degree expander Cayley ....

Alon, N., Milman, V.D.:  1 , isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory { B 38 (1985), 73-88.

....is to further develop the framework in which the interplay of Gamma r and Exp s can be studied. Hadamard Schur theory was discussed in Section 3. Perron Frobenius theory, graph partitioning by eigenvectors (e.g. 32, 33] and work regarding the second largest eigenvalue of a graph (e.g. [1, 6]) form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by Meyer may offer conceptual support in its focus on uncoupling Markov chains [26, 27] There are also papers which address the topic of matrix structure when the subdominant ....

N. Alon and V.D. Milman. 1 , Isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38:73--88, 1985.

....Esize(G) n X j=1 c j . Recall also that # denotes the maximum degree of a vertex. Given a set of vertices S, we denote by #(S) the set of edges with one endpoint in S and the other in V S. We make use of the following elementary result, where the lower bound is due to Alon and Milman [1] and the upper bound is due to Juvan and Mohar [24] Lemma 3.1. Let S # V be a subset of the vertices of a graph G. Then # 2 (Q) S V S n # #(S) ## n (Q) S V S n . # Theorem 3.2. The envelope size of a symmetric matrix A canbeboundedin terms of the eigenvalues of the ....

N. Alon and V. Milman, # 1 , isoperimetric inequalities for graphs, and superconcentrators,J. Combin. Theory Ser. B, 38 (1985), pp. 73--88.

.... manifolds, Alon [Alo86] and Sinclair and Jerrum [SJ89] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a cut with a good ratio of cut edges to separated vertices (see also [AM85, Fil91, DS91, Mih89, Moh89]) Around the same time, improvements in algorithms for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation of eigenvectors practical [PSS82, Sim91] In the next few years, a wealth of experimental work demonstrated that spectral partitioning methods work well ....

.... Mihail [Mih89] demonstrates that one can obtain a good ratio cut from any vector with small Rayleigh quotient that is perpendicular to the all ones s vector (although this is not explicitly stated in her work) In the full version of this paper, we present a new proof of Mihail s theorem (see also [AM85, Fil91, DS91, Mih89] and [Moh89] for a tighter bound) Theorem 3 (Mihail) Let G = V; E) be a graph on n nodes of maximum degree Delta, let L(G) be its Laplacian matrix, and let OE be its isoperimetric number. For any vector x 2 R n such that P n i=1 x i = 0, x T L(G) x x T x OE 2 2 Delta : ....

N. Alon and V. Milman. 1 , isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory Series B, 38:73--88, 1985.

....the motivation for proving Corollary 7 (for Pi n ) We were dealing here only with applications to functional analysis and convexity. There are many applications to other areas which we shall not expend on. There are applications to graph theory (see e.g. the construction of expander graphs in [1]) to other combinatorial questions, computer science, mathematical physics and probability (in particular to estimating the tail behavior of random variables of the form k P ffl i X i k for independent vector valued random variables fX i g) 57] contains many applications of the material of ....

N. Alon and V. D. Milman, 1 ; isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985) 73--88.

....exciting and extensive body of research, developed mainly by mathematicians intrigued by this computer science challenge. Most of this work was guided by the sufficient 1 condition for the expansion of (infinite families of constant degree regular) graphs discovered by Tanner [Tan84] see also [AM85] the second largest eigenvalue of the adjacency matrix should be strictly smaller than the degree. This naturally lead researchers to consider algebraic constructions, where this eigenvalue can be estimated. The celebrated sequence of papers [Mar73, GG81, AM85, AGM87, JM87, LPS88, Mar88, Mor94] ....

....by Tanner [Tan84] see also [AM85] the second largest eigenvalue of the adjacency matrix should be strictly smaller than the degree. This naturally lead researchers to consider algebraic constructions, where this eigenvalue can be estimated. The celebrated sequence of papers [Mar73, GG81, AM85, AGM87, JM87, LPS88, Mar88, Mor94] provided such constant degree expanders. All these graphs are extremely simple to describe: given the name of a vertex (in binary) its neighbors can be computed in polynomial time (or even logarithmic space) This level of explicitness is essential for many of ....

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N. Alon and V. D. Milman.  1 ; isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1):73--88, 1985.

....jffi(A; B)j=jAj, where A is the smaller of the two subsets. Donath and Hoffman [29] were the earliest to use spectral methods to partition graphs in the context of circuit layout. Spectral partitioning algorithms were considered and analyzed by Barnes [9] Boppana [12] Alon, Galil, and Milman [2], Mohar [80] and many others. Aspvall and Gilbert [6] used eigenvectors of the adjacency matrix to color the vertices of a graph. Pothen, Simon, and Liou [89] used the spectral partitioning algorithm to compute separators for parallel computing, proved additional lower bounds on separators, and ....

N. Alon and V. Milman, 1 , isoperimetric inequalities for graphs, and superconcentrators, J. of Combin. Theory, Series B, 38 (1985), pp. 73 -- 88.

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N. Alon and V. D. Milman. 1 , isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38:73--88, 1985.

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Alon, N., and Milman, V. D.  1 ; isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38, 1 (1985), 73-88.

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N. Alon and V.D. Milman. 1 , isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory Series B, 38:73--88, 1985.

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N. Alon and V.D. Milman. 1 ; isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory Series B, 38:73--88, 1985.

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N. Alon and V. D. Milman. isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1):73--88, 1985.

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N. Alon, and V.D. Milman, 1 , isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B, 38 (1985) 73 - 88.

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N. Alon and V. D. Milman. # 1 , isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory, Ser. B, 38:73--88, 1985.

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N. Alon and V. Milman.  1 , isoperimetric inequalities for graphs, and superconcentrators. J. of Combin. Theory, B(38):73-88, 1985.

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Alon, N. and Milman, V. D. (1985) 1 , isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B 38 73-88.

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N. Alon, V. D. Milman,  1 , isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory, Ser. B 38 (1985), 73-88.

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N. Alon, V. D. Milman, 1 , isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory, Ser. B 38 (1985), 73-88.

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