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80
Models of Random Regular Graphs
 IN SURVEYS IN COMBINATORICS
, 1999
"... In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular g ..."
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Cited by 215 (33 self)
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In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides.
A proof of Alon’s second eigenvalue conjecture
, 2003
"... A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 ..."
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Cited by 166 (1 self)
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A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 +ɛwith probability 1 − O(n−τ), where τ = ⌈ � √ d − 1+1 � /2⌉−1. We also show that this probability is at most 1 − c/nτ ′, for a constant c and a τ ′ that is either τ or τ +1 (“more often ” τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1
Undirected STConnectivity in LogSpace
, 2004
"... We present a deterministic, logspace algorithm that solves stconnectivity in undirected graphs. The previous bound on the space complexity of undirected stconnectivity was log 4/3 (·) obtained by Armoni, TaShma, Wigderson and Zhou [ATSWZ00]. As undirected stconnectivity is complete for the clas ..."
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Cited by 162 (3 self)
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We present a deterministic, logspace algorithm that solves stconnectivity in undirected graphs. The previous bound on the space complexity of undirected stconnectivity was log 4/3 (·) obtained by Armoni, TaShma, Wigderson and Zhou [ATSWZ00]. As undirected stconnectivity is complete for the class of problems solvable by symmetric, nondeterministic, logspace computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic logspace computations). Our algorithm also implies logspace constructible universaltraversal sequences for graphs with restricted labelling and logspace constructible universalexploration sequences for general graphs.
Entropy waves, the zigzag graph product, and new constantdegree expanders
, 2002
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Distributed Construction of Random Expander Networks
 In IEEE Infocom
, 2003
"... We present a novel distributed algorithm for constructing random overlay networks that are composed of d Hamilton cycles. The protocol is completely decentralized as no globallyknown server is required. The constructed topologies are expanders with O(log d n) diameter with high probability. ..."
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Cited by 103 (0 self)
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We present a novel distributed algorithm for constructing random overlay networks that are composed of d Hamilton cycles. The protocol is completely decentralized as no globallyknown server is required. The constructed topologies are expanders with O(log d n) diameter with high probability.
A spectral technique for coloring random 3colorable graphs
 SIAM Journal on Computing
, 1994
"... Abstract. Let G3n,p,3 be a random 3colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We des ..."
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Cited by 91 (2 self)
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Abstract. Let G3n,p,3 be a random 3colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomialtime algorithm that finds a proper 3coloring of G3n,p,3 with high probability, whenever p ≥ c/n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p ≥ polylog(n)/n [J. Algorithms, 19 (1995), pp. 204–234]. The algorithm can be extended to produce optimal kcolorings of random kcolorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c.
Uniform distribution of Heegner points
 Invent. math
"... Let E be a (modular!) elliptic curve over Q, of conductor N. Let K denote an imaginary quadratic field of discriminant D, with (N, D) = 1. If p is a prime, then there exists a unique Zpextension K∞/K such that Gal(K/Q) acts nontrivially on Gal(K∞/K). The field K ∞ is called the anticyclotomic Zpe ..."
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Cited by 52 (5 self)
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Let E be a (modular!) elliptic curve over Q, of conductor N. Let K denote an imaginary quadratic field of discriminant D, with (N, D) = 1. If p is a prime, then there exists a unique Zpextension K∞/K such that Gal(K/Q) acts nontrivially on Gal(K∞/K). The field K ∞ is called the anticyclotomic Zpextension of K. Let E(K∞) denote the MordellWeil group of E over K∞. Then a fundamental
Ramanujan graphs
 J. Ramanujan Math. Soc
, 1965
"... In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branc ..."
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Cited by 49 (0 self)
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In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The purpose of this survey is to unify some of the recent developments and expose certain open problems in the area. This survey is by no means an exhaustive one and demonstrates a highly numbertheoretic bias. For more comprehensive surveys, we refer the reader to [27], [9] or [13]. For a more uptodate survey highlighting the connection between graph theory and automorphic representations, we refer the reader to Winnie Li’s recent survey article [11]. A graph X is a triple consisting of a vertex set V = V(X),anedge set E = E(X) and a map that associates to each edge two vertices (not necessarily distinct) called its endpoints. Aloop is an edge whose endpoints are equal.
Eigenvalues and Expansion of Regular Graphs
 Journal of the ACM
, 1995
"... The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kr ..."
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Cited by 49 (1 self)
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The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kregular Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately k=2. Moreover, we construct a family of kregular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k=2. This shows that k=2 is the best bound one can obtain using the second eigenvalue method. We also show an upper bound of roughly 1 + p k \Gamma 1 on the average degree of linearsized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2 p k \Gamma 1. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (resp. extrovert graphs) of smaller size (resp. degree) th...
Lifts, Discrepancy and Nearly Optimal Spectral Gaps
"... Let G be a graph on n vertices. A 2lift of G is a graph H on 2n vertices, with a covering map : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n \new" eigenvalues. We conjecture that every dregular graph has a 2lift such that all new ..."
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Cited by 48 (4 self)
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Let G be a graph on n vertices. A 2lift of G is a graph H on 2n vertices, with a covering map : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n \new" eigenvalues. We conjecture that every dregular graph has a 2lift such that all new eigenvalues are in the range [ 2 d 1; 2 d 1] (If true, this is tight , e.g. by the AlonBoppana bound). Here we show that every graph of maximal degree d has a 2lift such that all \new" eigenvalues are in the range [ c d; c d] for some constant c.