## Pseudo-Random Graphs

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Venue: | IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15 |

Citations: | 46 - 15 self |

### BibTeX

@INPROCEEDINGS{Krivelevich_pseudo-randomgraphs,

author = {Michael Krivelevich and Benny Sudakov},

title = {Pseudo-Random Graphs},

booktitle = {IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15},

year = {},

pages = {199--262},

publisher = {Springer}

}

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### Citations

11412 |
Computers and Intractability. A Guide to the Theory of NP-Completeness. edited by W.H
- Garey, Johnson
- 1997
(Show Context)
Citation Context ...finding a cut of maximum size in G. Let f(G) be the size of the maximum cut in G. MAX CUT is one of the most natural combinatorial optimization problems. It is well known that this problem is NP-hard =-=[45]-=-. Therefore it is useful to have bounds on f(G) based on other parameters of the graph, that can be computed efficiently. Here we describe two such folklore results. First, consider a random partition... |

2067 |
The Theory of Error-Correcting Codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ...raph of the additive group Z3k 2 with respect to the generating set S = U0 + U1, where U0 = { (w0,w3 0 ,w5 0 ) | w0 } ∈ W0 and U1 is defined similarly. A well known fact from Coding Theory (see e.g., =-=[66]-=-), which can be proved using the Vandermonde determinant, is that every set of six distinct vectors in U0 ∪ U1 is linearly independent over GF(2). In particular all the vectors in U0 + U1 are distinct... |

2008 | On random graphs
- Erdös, Rényi
- 1959
(Show Context)
Citation Context ...roof of Theorem 4.23, referring the reader to [42]. One of the most important events in the study of random graphs was the discovery of the sudden appearance of the giant component by Erdős and Rényi =-=[33]-=-. They proved that all connected components of G(n,c/n) with 0 < c < 1 are almost surely trees or unicyclic and have size O(log n). On the other hand, if c > 1, then G(n,c/n) contains almost surely a ... |

753 | Random Graphs
- Janson, ÃLuczak, et al.
- 2000
(Show Context)
Citation Context ...bsection is to state all necessary definitions and results on random graphs. We certainly do not intend to be comprehensive here, instead referring the reader to two monographs on random graphs [20], =-=[49]-=-, devoted entirely to the subject and presenting a very detailed picture of the current research in this area. A random graph G(n,p) is a probability space of all labeled graphs on n vertices {1,... ,... |

561 |
Finite Fields
- Lidl, Niederreiter
- 1983
(Show Context)
Citation Context ...k = 2, when it is the Paley graph. Let χ be a nontrivial additive character of GF(q) and consider the Gauss sum ∑ y∈GF(q) χ(yk ). Using the classical bound | ∑ y∈GF(q) χ(yk )| ≤ (k − 1)q1/2 (see e.g. =-=[63]-=-) and the above connection between characters and eigenvalues we can conclude that the second largest eigenvalue of our graph Γ is bounded by O(q 1/2 ). 9. Next we present a surprising construction ob... |

440 | Algebraic Graph Theory
- Godsil, Royle
- 2001
(Show Context)
Citation Context ...orm, can be found in many textbooks in Linear Algebra. Readers more inclined to consult combinatorial books can find it for example in a recent monograph of Godsil and Royle on Algebraic Graph Theory =-=[46]-=-. We now prove a well known theorem (see its variant, e.g., in Chapter 9, [18]) bridging between graph spectra and edge distribution. Theorem 2.11 Let G be a d-regular graph on n vertices. Let d = λ1 ... |

312 |
Distance-Regular Graphs
- Brouwer, Cohen, et al.
- 1989
(Show Context)
Citation Context .... For every q which is an odd power of 2, the incidence graph of the generalized 4-gon has a polarity. The corresponding polarity graph is a (q + 1)-regular graph with q 3 + q 2 + q + 1 vertices. See =-=[23]-=-, [62] for more details. This graph contains no cycle of length 6 and it is not difficult to compute its eigenvalues (they can be derived, for example, from the eigenvalues of the corresponding bipart... |

216 |
Explicit group-theoretical constructions of combinatorial schemes and their application to desighn of expanders and concentrators, Probl
- Margulis
- 1988
(Show Context)
Citation Context ... ≤ 2h + 1 and that the second eigenvalue of G is bounded by O(2 k ). 11. Now we describe the celebrated expander graphs constructed by Lubotzky, Phillips and Sarnak [65] and independently by Margulis =-=[68]-=-. Let p and q be unequal primes, both congruent to 1 modulo 4 and such that p is a quadratic residue modulo q. As usual denote by PSL(2,q) the factor group of the group of two by two matrices over GF(... |

165 | Thresholds of Graph Properties, and the k-sat Problem
- Friedgut, Sharp
- 1999
(Show Context)
Citation Context ...ar subjects in the study of random graphs is proving sharpness of thresholds for various combinatorial properties. This direction of research was spurred by a powerful 40theorem of Friedgut-Bourgain =-=[37]-=-, providing a sufficient condition for the sharpness of a threshold. The authors together with Vu apply this theorem in [60] to show sharpness of graph connectivity, sometimes also called network reli... |

157 | Models of random regular graphs
- Wormald
- 1999
(Show Context)
Citation Context ...−|E(G)| . We will occasionally mention also the probability space Gn,d, this is the probability space of all d-regular graphs on n vertices endowed with the uniform measure, see the survey of Wormald =-=[83]-=- for more background. We also say that a graph property A holds almost surely, or a.s. for brevity, in G(n,p) (Gn,d) if the probability that G(n,p) (Gn,d) has A tends to one as the number of vertices ... |

102 |
The Probabilistic Method, 2nd ed
- Alon, Spencer
- 2000
(Show Context)
Citation Context ...W) are binomially distributed random variables with parameters ( u 2 and p, respectively. Applying standard Chernoff-type estimates on the tails of the binomial distribution (see, e.g., Appendix A of =-=[18]-=-) and then the union bound, one gets the desired inequalities. It is very instructive to notice that we get less and less control over the edge distribution as the set size becomes smaller. For exampl... |

96 | The algorithmic aspects of the Regularity Lemma
- Alon, Duke, et al.
- 1994
(Show Context)
Citation Context ...constructions are jumbled. Also, it can find algorithmic applications, for example, a very similar approach has been used by Alon, Duke, Lefmann, Rödl and Yuster in their Algorithmic Regularity Lemma =-=[9]-=-. As observed by Thomason, the minimum degree condition of Theorem 2.4 can be dropped if we require that every pair of vertices has (1 + o(1))np 2 common neighbors. One cannot however weaken the condi... |

96 | A proof of alon's second eigenvalue conjecture
- Friedman
- 2003
(Show Context)
Citation Context ...from those of the binomial random graph G(n,p),p = d/n. For example, they are almost surely connected. The spectrum of Gn,d for a fixed d was studied in [38] by Friedman, Kahn and Szemerédi. Friedman =-=[39]-=- proved that for constant d the second largest eigenvalue of a random d-regular graph is λ = (1 + o(1))2 √ d − 1. The approach of Kahn and Szemerédi gives only O( √ d) bound on λ but continues to work... |

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

77 | Restricted colorings of graphs
- Alon
- 1993
(Show Context)
Citation Context ...ery vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). The study of this parameter received a considerable amount of attention in recent years, see, e.g., =-=[2]-=-, [57] for two surveys. Note that from the definition it follows immediately that χl(G) ≥ χ(G) and it is known that the gap between these two parameters can be arbitrarily large. The list-chromatic nu... |

74 |
Cycles of even length in graphs
- Bondy, Simonovits
- 1974
(Show Context)
Citation Context ...,d,λ)-graph contains a cycle of length 2k + 1. As shown by Example 10 of the previous section this result is tight. It is worth mentioning here that it follows from the result of Bondy and Simonovits =-=[22]-=- that any d-regular graph with d ≫ n 1/k contains a cycle of length 2k. Here we do not need to make any assumption about the second eigenvalue λ. This bound is known to be tight for k = 2,3,5 (see Exa... |

74 |
Hamiltonian circuits in random graphs
- Pósa
- 1976
(Show Context)
Citation Context ...nd then G is Hamiltonian. λ ≤ (log log n) 2 1000log n(log log log n) d, The proof of Theorem 4.20 is quite involved technically. Its main instrument is the famous rotation-extension technique of Posa =-=[70]-=-, or rather a version of it developed by Komlós and Szemerédi in [56] to obtain the exact threshold for the appearance of a Hamilton cycle in the random graph G(n,p). We omit the proof details here, r... |

69 |
On the value of a random minimum spanning tree problem
- Frieze
- 1985
(Show Context)
Citation Context ...ith d/2 < |S| ≤ min{ρd, |V |/2} . mst(G) = (1 + o(1)) |V | ζ(3) , d where the o(1) term tends to 0 as d → ∞, and ζ(3) = ∑ ∞ i=1 i−3 = 1.202.... The above theorem extends a celebrated result of Frieze =-=[40]-=-, who proved it in the case of the complete graph G = Kn. Pseudo-random graphs supply easily the degree of edge expansion required by Theorem 4.27. We thus get: Corollary 4.28 Let G be an (n,d,λ)-grap... |

68 | Random graphs, 2 nd ed - Bollobás - 2001 |

61 |
On the second eigenvalue in random regular graphs
- FRIEDMAN, KAHN, et al.
- 1989
(Show Context)
Citation Context ...regular graphs have quite different properties from those of the binomial random graph G(n,p),p = d/n. For example, they are almost surely connected. The spectrum of Gn,d for a fixed d was studied in =-=[38]-=- by Friedman, Kahn and Szemerédi. Friedman [39] proved that for constant d the second largest eigenvalue of a random d-regular graph is λ = (1 + o(1))2 √ d − 1. The approach of Kahn and Szemerédi give... |

57 |
Proof of a conjecture of Erdős, Combinatorial Theory and its Applications Vol
- Hajnal, Szemerédi
- 1970
(Show Context)
Citation Context ...ally assuming that H is fixed while the order n of G grows. In many cases such conditions are formulated in terms of the minimum degree of G. For example, the classical result of Hajnal and Szemerédi =-=[47]-=- asserts that if the minimum degree δ(G) satisfies δ(G) ≥ (1 − 1 r )n, then G contains ⌊n/r⌋ vertex disjoint copies of Kr. The statement of this theorem is easily seen to be tight. 34It turns our tha... |

54 | A note on hamiltonian circuits
- Chvátal, Erdös
- 1972
(Show Context)
Citation Context ...miltonian. d − 36 λ2 d ≥ λn d + λ , Proof. According to Theorem 4.1 G is (d − 36λ2 /d)-vertex-connected. Also, α(G) ≤ λn/(d + λ), as stated in Proposition 4.5. Finally, a theorem of Chvátal and Erdős =-=[29]-=- asserts that if the vertexconnectivity of a graph G is at least as large as its independence number, then G is Hamiltonian. ✷ The Chvátal-Erdős Theorem has also been used by Thomason in [79], who pro... |

52 |
Strongly regular graphs, partial geometries and partially balanced designs
- Bose
- 1963
(Show Context)
Citation Context ...e examples of strongly regular graph are the pentagon C5 that has parameters (5,2,0,1), and the Petersen graph whose parameters are (10,3,0,1). Strongly regular graphs were introduced by Bose in 1963 =-=[21]-=- who also pointed out their tight connections with finite geometries. As follows from the definition, strongly regular graphs are highly regular structures, and one can safely predict that algebraic m... |

51 |
Probabilistic characteristics of graphs with large connectivity, Problems Info. Transmission 10
- Margulis
- 1977
(Show Context)
Citation Context ...es, maximum degree di and it is ki-edge-connected. If ki ln ni lim i→∞ di = ∞, then the family (Gi) ∞ i=1 has a sharp connectivity threshold. The above theorem extends a celebrated result of Margulis =-=[67]-=- on network reliability (Margulis’ result applies to the case where the critical probability is a constant). Since (n,d,λ) graphs are d(1 − o(1))-connected as long as λ = o(d) by Theorem 4.1, we immed... |

50 |
Coloring the vertices of a graph in prescribed colors (in
- Vizing
- 1976
(Show Context)
Citation Context ...Pq) = √ q ≪ q/log q. Therefore the bound in Corollary 4.7 is best possible. A more complicated quantity related to the chromatic number is the list-chromatic number χl(G) of G, introduced in [34] and =-=[82]-=-. This is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). The st... |

49 | Quasi-random graphs
- Chung, Graham, et al.
(Show Context)
Citation Context ...ired to hold for all graphs on l vertices simultaneously. Let us switch now to the case of vanishing edge density p(n) = o(1). This case has been treated in two very recent papers of Chung and Graham =-=[25]-=- and of Kohayakawa, Rödl and Sissokho [50]. Here the picture becomes significantly more complicated compared to the dense case. In particular, 8there exist graphs with very balanced edge distribution... |

46 |
Random Cayley graphs and expanders, Random Structures Algorithms
- ALON, ROICHMAN
- 1994
(Show Context)
Citation Context ...r for d ≥ clog5 n. Therefore to prove Hamiltonicity of X(G,S), by Theorem 4.20 it is enough to show that almost surely λ/d ≤ O(log n). This can be 38done by applying an approach of Alon and Roichman =-=[16]-=- for bounding the second eigenvalue of a random Cayley graph. We note that a well known conjecture claims that every connected Cayley graph is Hamiltonian. If true the conjecture would easily imply th... |

46 | On the second eigenvalue of a graph - NILLI - 1991 |

44 | Norm-graphs: variations and applications
- Alon, Rónyai, et al.
- 1999
(Show Context)
Citation Context ...e. The graphs G p,q have very good expansion properties and have numerous applications in Combinatorics and Theoretical Computer Science. 12. The projective norm graphs NGp,t have been constructed in =-=[17]-=-, modifying an earlier construction given in [52]. These graphs are not Cayley graphs, but as one will immediately see, their construction has a similar flavor. The construction is the following. Let ... |

44 |
Pseudo-random graphs
- Thomason
- 1987
(Show Context)
Citation Context ...e introduced. Although first examples and applications of pseudo-random graphs appeared very long time ago, it was Andrew Thomason who launched systematic research on this subject with his two papers =-=[79]-=-, [80] in the mid-eighties. Thomason introduced the notion of jumbled graphs, enabling to measure in quantitative terms the similarity between the edge distributions of pseudo-random and ∗ Department ... |

44 | Percolation on finite graphs and isoperimetric inequalities, Annals of Probability 32
- Alon, Benjamini, et al.
- 2004
(Show Context)
Citation Context ...d all remaining vertices are in components of size at least γn. These components are easily shown to merge quickly into one giant component of a linear size. The detail can be found in [43] (see also =-=[7]-=- for some related results). One of the recent most popular subjects in the study of random graphs is proving sharpness of thresholds for various combinatorial properties. This direction of research wa... |

42 |
Random graphs, strongly regular graphs and pseudo-random graphs
- Thomason
- 1987
(Show Context)
Citation Context ... is (p,7 √ ( nα/η/(1 − η))-jumbled. Moreover G contains a subset U ⊆ V (G) of size |U| ≥ 1 − 380 n(1−η) 2 ) w n such that the induced subgraph G[U] is (p,ωα)-jumbled. Thomason also describes in [79], =-=[80]-=- several properties of jumbled graphs. We will not discuss these results in details here as we will mostly adopt a different approach to pseudo-randomness. Occasionally however we will compare some of... |

41 |
personal communication
- Alon
- 2006
(Show Context)
Citation Context ...f a graph G is (p,α)-jumbled and p s n ≫ 42αs 2 then the number of induced subgraphs of G which are isomorphic to H is (1 + o(1))ps (1 − p) (s 2)−r ns /|Aut(H)|. Here we present a result of Noga Alon =-=[6]-=- that proves that every large subset of the set of vertices of (n,d,λ)-graph contains the ”correct” number of copies of any fixed sparse graph. An additional advantage of this result is that its asser... |

40 |
The eigenvalues of random symmetric matrices, Combinatorica 1
- Füredi, Komlós
- 1981
(Show Context)
Citation Context ...andom graph with edge probability p. If p satisfies pn/log n → ∞ and (1 − p)n log n → ∞, then almost surely all the degrees of G are equal to (1 + o(1))np. Moreover it was proved by Füredi and Komlós =-=[44]-=- that the largest eigenvalue of G is a.s. (1+o(1))np and that λ(G) ≤ (2+o(1)) √ p(1 − p)n. They stated this result only for constant p but their proof shows that λ(G) ≤ O( √ np) also when p ≥ poly log... |

39 | Explicit Ramsey graphs and orthonormal labelings, Electron - Alon - 1994 |

39 | The nonexistence of certain generalized polygons - Feit, Higman - 1964 |

38 |
Explicit concentrators from generalized n-gons
- Tanner
- 1984
(Show Context)
Citation Context ...graph contains no cycle of length 6 and it is not difficult to compute its eigenvalues (they can be derived, for example, from the eigenvalues of the corresponding bipartite incidence graph, given in =-=[78]-=-). Indeed, all the eigenvalues, besides the trivial one (which is q + 1) are either 0 or √ 2q or − √ 2q. Similarly, for every q which is an odd power of 3, the incidence graph of the generalized 6-gon... |

37 |
Combinatorial Problems and Exercises, 2nd edition
- Lovasz
- 1993
(Show Context)
Citation Context ...y adding o(|E(G0|) edges, G still satisfies DISC. On the other hand, G contains a star S of size n 0.8 with a center at v ∗ , and hence λ1(G) ≥ λ1(S) = √ n 0.8 − 1 ≫ |E(G)|/n (see, e.g. Chapter 11 of =-=[64]-=- for the relevant proofs). This solves an open question from [25]. ✷ 9The Erdős-Rényi graph from the next section is easily seen to satisfy EIG, but fails to satisfy CIRCUIT(4). Chung and Graham prov... |

35 |
Strongly regular graphs and partial geometries
- Brouwer, Lint
- 1984
(Show Context)
Citation Context ... extremely useful in their study. We do not intend to provide any systematic coverage of this fascinating concept here, addressing the reader to the vast literature on the subject instead (see, e.g., =-=[24]-=-). Our aim here is to calculate the eigenvalues of strongly regular graphs and then to connect them with pseudo-randomness, relying on results from the previous subsection. 15Proposition 2.12 Let G b... |

34 |
Limit distributions for the existence of Hamilton circuits in a random graph
- Komlós, Szemerédi
- 1983
(Show Context)
Citation Context ...d, The proof of Theorem 4.20 is quite involved technically. Its main instrument is the famous rotation-extension technique of Posa [70], or rather a version of it developed by Komlós and Szemerédi in =-=[56]-=- to obtain the exact threshold for the appearance of a Hamilton cycle in the random graph G(n,p). We omit the proof details here, referring the reader to [58]. For reasonably good pseudo-random graphs... |

31 | Approximating the independence number via the #-function - Alon, Kahale - 1998 |

31 | Constructive bounds for a Ramsey-type problem
- Alon, Krivelevich
- 1997
(Show Context)
Citation Context ...t most two possible choices of xt. Actually using more complicated computation, which we omit, one can determine the exact number of vertices with loops. The eigenvalues of G are easy to compute (see =-=[11]-=-). Indeed, let A be the adjacency matrix of G. Then, by the properties of PG(q,t), A 2 = AA T = µJ + (dq,t − µ)I, where µ = ( q t−1 − 1 ) / ( q − 1 ) , J is the all one matrix and I is the identity ma... |

30 | Approximating the independence number via the θ-function
- Alon, Kahale
- 1998
(Show Context)
Citation Context ... the characters of Z3k 2 it was proved in [3] that the second eigenvalue of Gn is bounded by λ ≤ 9 · 2k + 3 · 2k/2 + 1/4. 10. The construction above can be extended in the obvious way as mentioned in =-=[10]-=-. Let h ≥ 1 and suppose that k is an integer such that 2 k −1 is not divisible by 4h+3. Let W0 be the set of all nonzero elements α ∈ GF(2 k ) so that the leftmost bit in the binary representation of ... |

28 | Random regular graphs of non-constant degree: connectivity and Hamilton cycles
- Cooper, Frieze, et al.
- 1993
(Show Context)
Citation Context ...te this parameter by κ(G). For random graphs it is well known (see, e.g., [20]) that the vertex-connectivity is almost surely the same as the minimum degree. Recently it was also proved (see [61] and =-=[30]-=-) that random d-regular graphs are d-vertex-connected. For (n,d,λ)-graphs it is easy to show the following. Theorem 4.1 Let G be an (n,d,λ)-graph with d ≤ n/2. Then the vertex-connectivity of G satisf... |

27 |
Norm-graphs and bipartite Turán numbers, Combinatorica 16
- Kollár, Rónyai, et al.
- 1996
(Show Context)
Citation Context ...erties and have numerous applications in Combinatorics and Theoretical Computer Science. 12. The projective norm graphs NGp,t have been constructed in [17], modifying an earlier construction given in =-=[52]-=-. These graphs are not Cayley graphs, but as one will immediately see, their construction has a similar flavor. The construction is the following. Let t > 2 be an integer, let p be a prime, let GF(p) ... |

27 |
Random regular graphs of high degree, Random Structures and Algorithms 18
- Krivelevich, Sudakov, et al.
- 2001
(Show Context)
Citation Context ...The approach of Kahn and Szemerédi gives only O( √ d) bound on λ but continues to work also when d is small power of n. The case d ≫ n 1/2 was recently studied by Krivelevich, Sudakov, Vu and Wormald =-=[61]-=-. They proved that in this case for any two vertices u,v ∈ Gn,d almost surely ∣ codeg(u,v) − d 2 /n ∣ ∣ < Cd 3 /n 2 + 6d √ log n/ √ n, where C is some constant and codeg(u,v) is the number of common n... |

27 |
A survey of two-graphs’, in: Colloquio Internazionale sulle Teorie Combinatoire, Accademia Nazionale dei Lincei
- Seidel
- 1976
(Show Context)
Citation Context ...f the space into two sets P and N, where |P | = k. Two vertices x and y of the graph G are adjacent if x − y is parallel to a line in P. This example is due to Delsarte and Goethals and to Turyn (see =-=[72]-=-). It is easy to check that G is strongly regular with parameters ( k(q − 1),(k − 1)(k − 2)+q −2,k(k −1) ) . Therefore its eigenvalues, besides the trivial one are −k and q −k. Thus if k is sufficient... |

24 |
New trends in the theory of graph colorings: choosability and list coloring, Contemporary Trends
- Tuza, Voigt
(Show Context)
Citation Context ...ertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). The study of this parameter received a considerable amount of attention in recent years, see, e.g., [2], =-=[57]-=- for two surveys. Note that from the definition it follows immediately that χl(G) ≥ χ(G) and it is known that the gap between these two parameters can be arbitrarily large. The list-chromatic number o... |

24 | Sparse pseudo-random graphs are Hamiltonian
- Krivelevich, Sudakov
(Show Context)
Citation Context ...condition starts working when d = Ω(n2/3 ). 37One can however prove a much stronger asymptotical result, using more sophisticated tools for assuring Hamiltonicity. The authors prove such a result in =-=[58]-=-: Theorem 4.20 [58] Let G be an (n,d,λ)-graph. If n is large enough and then G is Hamiltonian. λ ≤ (log log n) 2 1000log n(log log log n) d, The proof of Theorem 4.20 is quite involved technically. It... |

23 | The Blow-up lemma - Komlós - 1999 |