F.R.K. Chung, Diameters and eigenvalues, Journal of American Mathematical Society 2 (1989) No. 2, 187-196.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cardinality in Theorem 14 is tight, up to the dependence on d and . We require an upper estimate for the diameter of an n point expander. It is well known that the diameter is O(log n) but here we will be a little bit more accurate. We need the following bound on the diameter of expander graphs [17]: Proposition 43. Let G = V; E) be a n vertex, d regular graph. Denote = G) Then the diameter of G is at most log 1= n 1. Proposition 44. Let G = V; E) be a n vertex, d regular graph, d 3, and set = G) Then, there is an absolute constant C 0 such that for any 1, R 2 ....

F. R. K. Chung. Diameters and eigenvalues. J. Amer. Math. Soc., 2(2):187-196, 1989.

....ned for every t 2 ZN . Clearly, f A (0) jAj. The parameter (A) max t6=0 jf A (t)j=jAj gives some measure of the randomness of the set A; the smaller it is, the more random A is. This parameter has a variety of applications in Additive Number Theory (e.g. Ruzsa, AIKPS] Graph Theory (e.g. [Chung]) and Complexity Theory (e.g. ABK , GKS] The connection of this parameter with our problem stems from the fact that there are many cancelations in the sum of unit vectors that are almost uniformly distributed on the unit cycle. An easy calculation (which for completeness is given in the ....

F. R. K. Chung, Diameters and eigenvalues, J. AMS 2, pp. 187-196, 1989.

....over the n th roots of unity. Letting n p2 1, where p is prime, we have that F2 p ( where a; is the root of an irreducible quadratic polynomial in Fp Ix] The key technical fact is a theorem of Katz on character sums: if is any mukiplicative character on Fp (a; hen Using this result, CMng [2] observes that the following generating set yields a Ramanujan graph. Let g be a generator for F, and define ai logg( i) Then the p 1 set S ai i=o generates Z nZ, and has size 2p 2 [S[ 2p. Katz s theorem, applied to the character (g) 0, shows that the Caylcy graph F(C,S) is Ramanujan. ....

F. Chung, "Diameters and eigenvalues," J. Amer. Math. Soc., Vol. 2, 1989, pp. 18%196.

....one may consult [2] 5] 6] 8] 10] 12] 13] 15] Other authors have considered as well combinatorial consequences of various results concerning the distribution of the values taken by a multiplicative character of a finite field on a coset of a certain subfield. See, for example, [3]. In the third section of the paper we provide an extension of the basic construction, the result of which will be, for any n 2, a class of codes C(q; n) with similar properties as C(q; 2) but only with an almost constant weight for their codewords. Note that whenever we take off the first ....

Chung, F.R.K., Diameters and eigenvalues, Journal of the American Math.Soc., vol.2, no.2 (1989), 187-196

....[8] and Schwenk and Wilson [38] In this context, some of the recent work has been specially concerned with the study of metric parameters, such as the mean distance, diameter, radius, isoperimetric number, etc. See, for instance, the papers of Alon and Milman [1] Biggs [3] Chung et.al. 7] [6] , Van Dam and Haemers [11] Delorme and Sole [13] Kahale [31] Mohar [32] Quenell [36] Sarnak [39] and Garriga, Yebra, and the author [16] 19] 22] We must also mention here Haemers thesis [27] an account of which can be found in his recent paper [28] Somewhat surprisingly, in ....

F.R.K. Chung, Diameter and eigenvalues, J. Amer. Math. Soc. 2, No. 2 (1989) 187--196.

.... networks, it is required that the underlying h regular graphs have su#ciently many nodes, and it is desirable not only to keep h as small as possible (in order to reduce the complexity of the network) but also to minimize the diameter (so that information can be transmitted e#ciently) In [2], a construction of graphs with the above properties is proposed using the finite fields of the form F q n . Namely, for any prime q and any integer n # 2 with q (n 1) 2 , the construction produces q regular graphs G(q, n) with q n 1 nodes and with diameter D (G(q, n) # 2n 4n ....

....the construction produces q regular graphs G(q, n) with q n 1 nodes and with diameter D (G(q, n) # 2n 4n log n log q 2 log(n 1) 1) In [11] a more flexible construction has been proposed that produces h regular graphs for any h # q 1 2 # , # 0. The inequality (1) of [2] is based on bounds for very short character sums considered in [1, 7] while the result of [11] is based on bounds for even shorter sums in [10] All of these estimates are derived from the celebrated Weil bound. There are several other similar constructions and bounds for character sums; see [3, ....

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F. R. K. Chung, `Diameters and eigenvalues', J. Amer. Math. Soc. 2 (1989), 187--196.

....Note that if we set = max i6=n j i j, then graphs with small relative to k are not only good expanders, but also have high chromatic number. Finally, it is worth mentioning that the second eigenvalue of the adjacency matrix also bounds the diameter of the graph as a result of Chung s inequality [C] diameter(X) log(n Gamma 1) log(k= For a more complete set of references and a proper discussion of these results we refer to [B, L, S] It is known that certain families of Cayley graphs for SL 2 (q) are expanders. In particular, Lubotzky [L] shows that the uniform bound 1 ( GammanH) ....

F. Chung. Diameters and eigenvalues. Jour. of A.M.S., 2:187-200, 1989.

....one may consult [2] 5] 6] 8] 10] 12] 13] 15] Other authors have considered as well combinatorial consequences of various results concerning the distribution of the values taken by a multiplicative character of a finite field on a coset of a certain subfield. See, for example, [3]. In the third section of the paper we provide an extension of the basic construction, the result of which will be, for any n 2, a class of codes C(q; n) with similar properties as C(q; 2) but only with an almost constant weight for their codewords. Note that whenever we take off the first ....

Chung, F.R.K., Diameters and eigenvalues, Journal of the American Math.Soc., vol.2, no.2 (1989), 187-196

....and Theoretical Computer Science, Volume 10, 1993; pp. 49 62. c flAmerican Mathematical Society. bound on the diameter in terms of the eigenvalue gap is not surprising. Such a bound first appeared in [3] where it was shown that, when G is k regular, D(G) 2 s 2k (k 0 1 ) log 2 n: 1) Chung [6] established that D(G) log(n 0 1) log(k= 1; 2) which beats Eq. 1 when is small. Eq. 2 was further improved in [7, 16, 17] where it was shown that D(G) cosh 01 (n 0 1) cosh 01 (k= 1: 3) For fixed n and k, the right hand side is small when is small. It is known, ....

F. R. K. Chung. Diameters and eigenvalues. J. Amer. Math. Soc., 2(2):187--196, 1989.

....where p is prime, we have that F # p 2 # = F p (#) where # is the root of an irreducible quadratic polynomial in F p [x] The key technical fact is a theorem of Katz on character sums: if is any multiplicative character on F p (#) then X i # Fp #(# i) # # p . Using this result, Chung [2] observes that the following generating set yields a Ramanujan graph. Let g be a generator for F # p 2 , and define a i = log g (# i) Then the set S = a i p 1 i=0 generates Z nZ, and has size 2p 2 # S # 2p. Katz s theorem, applied to the character #(g) #, shows that the Cayley ....

F. Chung, "Diameters and eigenvalues," J. Amer. Math. Soc., Vol. 2, 1989, pp. 187--196.

....weaker questions about generators of finite fields. The first one is as follows. Question 1.1. For which pair (q; m) the multiplicative group F q m is generated by the line F q ff for every ff with F q m = F q (ff) This question also arises from several applications such as graph theory [Ch] and number theoretic algorithms [Le2] For a given ff with F q m = F q (ff) define the difference graph G(m; q; ff) to be the graph whose vertices are the elements of the multiplicative group F q m , where two elements fi 1 and fi 2 are connected if and only if fi 1 =fi 2 = ff a) for some ....

....of the multiplicative group F q m , where two elements fi 1 and fi 2 are connected if and only if fi 1 =fi 2 = ff a) for some a in the ground field F q . This is a regular graph of degree q, i.e. each vertex is connected to exactly q other vertices. The difference graph is studied in [Ch] and more generally in [Li] It is clear that the graph G(m; q; ff) Partially supported by NSF 1991 Mathematics Subject Classification Numbers: 11T24, 11T55 Typeset by A M S T E X 1 is connected if and only if F q m is generated by the line F q ff. Thus, the graph G(m; q; ff) is connected ....

[Article contains additional citation context not shown here]

F.R. Chung, Diameters and eigenvalues, J. Amer. Math. Soc., 2(1989), 187-196.

....Doob [8] and Schwenk and Wilson [38] In this context, some of the recent work has been specially concerned with the study of metric parameters, such as the mean distance, diameter, radius, isoperimetric number, etc. See, for instance, the papers of Alon and Milman [1] Biggs [3] Chung et.al. [6, 7], Van Dam and Haemers [11] Delorme and Sole [13] Kahale [31] Mohar [32] Quenell [36] Sarnak [39] and Garriga, Yebra, and the author [16, 19, 22] We must also mention here Haemers thesis [27] an account of which can be found in his recent paper [28] Somewhat surprisingly, in some of these ....

F.R.K. Chung, Diameter and eigenvalues, J. Amer. Math. Soc. 2, No. 2 (1989) 187--196.

....one may consult [2] 5] 6] 8] 10] 12] 13] 15] Other authors have considered as well combinatorial consequences of various results concerning the distribution of the values taken by a multiplicative character of a finite field on a coset of a certain subfield. See, for example, [3]. In the third section of the paper we provide an extension of the basic construction, the result of which will be, for any n 2, a class of codes C(q,n) with similar properties as C(q,2) but only with an almost constant weight for their codewords. Note that whenever we take o# the first row ....

Chung, F.R.K., Diameters and eigenvalues, Journal of the American Math.Soc., vol.2, no.2 (1989), 187-196

....of which relate directly to finite fields including numerous problems related to orthomorphisms. We refer the reader to this list for a rich source of open problems. We close this section on combinatorics by raising several problems related to the construction of small diameter graphs, see Chung [11] and Cohen [16] In order that information can be transmitted efficiently in a network, the underlying graphs of various communications networks are often required to have small diameters. Consider the following quite general setting. For a polynomial f over F q let K = F q [x] f(x) be an n ....

....a c ) a i 2 F q : If q is sufficiently large compared to n, the diameter D( Gamma) of Gamma is generally small and in fact close to the lower bound D( Gamma) n 1, see Cohen [16] for details. Various special cases of this construction have been considered in the literature; for example Chung [11] studied the case when f is irreducible (and thus K is the finite field F q n ) Katz [33] considered the case when f is squarefree while Cohen [16] considered the case when f is tame (the characteristic p of F q does not divide the multiplicity of the roots of f ) The case when p = 2 is ....

[Article contains additional citation context not shown here]

F. R. K. Chung, `Diameters and eigenvalues', J. Amer. Math. Soc., 2(1989), 187--196.

....digraphs is not required to be bounded. We shall now describe an efficient construction of digraphs with an almost optimal separation exponent, and a degree polylogarithmic in the size of the digraph. The basis for our construction is an explicit definition of highly separable digraphs given by Chung [Chu] The construction of Chung is obtained as follows: Let F be a finite field say of characteristic 2, and f an irreducible polynomial of degree t. Let E be the extension field of F formed by adjoining a root w of f to F , and g a generator for E (the multiplicative group of invertible ....

....is not required to be bounded. We shall now describe an efficient construction of digraphs with an almost optimal separation exponent, and a degree polylogarithmic in the size of the digraph. The basis for our construction is an explicit definition of highly separable digraphs given by Chung [Chu] The construction of Chung is obtained as follows: Let F be a finite field say of characteristic 2, and f an irreducible polynomial of degree t. Let E be the extension field of F formed by adjoining a root w of f to F , and g a generator for E (the multiplicative group of invertible elements ....

[Article contains additional citation context not shown here]

F. R. K. Chung. Diameters and eigenvalues. Manuscript.

....for every t 2 ZN . Clearly, fA (0) jAj. The parameter (A) max t6=0 jf A (t)j=jAj gives some measure of the randomness of the set A; the smaller it is, the more random A is. This parameter has a variety of applications in Additive Number Theory (e.g. Ruzsa, AIKPS] Graph Theory (e.g. [Chung]) and Complexity Theory (e.g. ABK , GKS] The connection of this parameter with our problem stems from the fact that there are many cancelations in the sum of unit vectors that are almost uniformly distributed on the unit cycle. An easy calculation (which for completeness is given in the ....

F. R. K. Chung, Diameters and eigenvalues, J. AMS 2, pp. 187--196, 1989.

....October 1 4, 1991; pp. 398 404. c flIEEE, and on On the Second Eigenvalue and Linear Expansion of Regular Graphs , by Nabil Kahale, which appeared in DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 10, 1993; pp. 49 62. c flAmerican Mathematical Society. Chung [5] (see also [12] established that D(G) log(n Gamma 1) log(k= 1; 2) which beats Eq. 1 when is small. Eq. 2 was further improved in [6, 16] where it was shown that D(G) cosh Gamma1 (n Gamma 1) cosh Gamma1 (k= 1: 3) In this paper, we establish isoperimetric bounds ....

F. R. K. Chung. Diameters and eigenvalues. J. Amer. Math. Soc., 2(2):187--196, 1989.

.... Gamma )S ( Gamma 1 k Gamma1 S ( Gamma 1) 1 Gamma )S ( 1 k Gamma1 ) 1 Gamma )S ( Gamma S ( Gamma 1) 1 Gamma )S ( 1 k Gamma1 ) S ( Gamma S ( Gamma 1) 1 Gamma )S ( As a consequence, we get the estimate of F. R. Chung: Corollary 4. 10 ([12]) diam( Gamma) log(#( Gamma) Gamma 1) Gamma log(1 Gamma ) We remark that this estimate can be improved substantially from our present techniques, by using a better upper estimate for S ( We will leave this line of thought to the reader. We will see below, however, how to improve this ....

F. R. K. Chung,"Diameters and Eigenvalues," J. AMS 2 (1989), pp. 187-196.

....code of size k in the graph be. Two problems on the diameter have excited a great deal of attention since the 0 80s: ffl the ( Delta; D) graph problem: how large can be a graph of bounded degree and given diameter [1, 6, 5] ffl finding the best spectral upper bound on the diameter of a graph. [2, 3] This work is an attempt to generalize both philosophies. First, we study a function N (k; Delta; D) the largest size of a graph of degree at most Delta and given k diameter D. Observe that this is a very hard problem which comprises as special instances both Moore graphs and perfect codes. We ....

....that this is a very hard problem which comprises as special instances both Moore graphs and perfect codes. We begin a 3 D table collecting the sizes of the largest such graphs. No exact value with Delta 2 is known so far. Second we derive the natural analogues of the Chung et al. upper bounds [2, 3, 7] on the diameter. Combining the two types of argument we obtain some lower bounds on the spectral multiplicity of a kind which seems to be new. 2 Definitions and Notations All graphs considered are finite, connected, with vertex set V , simple, undirected, without loops or multiple edges. The ....

F.R.K. Chung, Diameter and eigenvalues, J. of the AMS 2 (1989) 187-196.

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F.R.K. Chung, Diameters and eigenvalues, Journal of American Mathematical Society 2 (1989) No. 2, 187-196.

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F. Chung, "Diameters and eigenvalues," J. Amer. Math. Soc., Vol. 2, 1989, pp. 187--196.

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F. Chung, "Diameters and eigenvalues", J. Amer. Math. Soc. 2 (1989) 187--200.

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F. R. K. Chung. Diameters and eigenvalues. J. Amer. Math. Soc., 2(2):187-196, 1989.

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F. R. K. Chung. Diameters and eigenvalues. J. Amer. Math. Soc. 2, 187--196 (1989).

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