Lifts, discrepancy and nearly optimal spectral gap

Combinatorica

Volumn 26, Issue 5, 2006, Pages 495-519

  Bilu, Yonatan ^a   Linial, Nathan ^b  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33847367664     PISSN: 02099683     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00493-006-0029-7     Document Type: Article

References (32)

1. M. AJTAI: Recursive construction for 3-regular expanders, Combinatorica 14(4) (1994), 379-416.
2. N. ALON: Personal communication.
3. N. ALON: Eigenvalues and expanders, Combinatorica 6(2) (1986), 83.96;
4. Theory of computing (Singer Island, Fla., 1984).
5. N. ALON: A parallel algorithmic version of the local lemma, Random Structures Algorithms 2(4) (1991), 367-378.
6. N. ALON, Z. Galil and V. D. Milman: Better expanders and superconcentrators, J.A lgorithms 8(3) (1987), 337-347.
7. N. ALON, O. GOLDREICH J. HÅSTAD and R. PERALTA: Simple constructions of almost k-wise independent random variables, in 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), pages 544.553. IEEE Comput. Soc. Press, Los Alamitos, CA, 1990.
8. 1, isoperimetric inequalities for graphs, and superconcentrators; J. Combin. Theory Ser. B 38(1) (1985), 73-88.
9. N. ALON and J. H. SPENCER: The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000. Withan appendix on the life and work of Paul Erdos.
10. A. AMIT and N. LINIAL: Random graphc overings I: General theory and graph connectivity; Combinatorica 22(1) (2002), 1-18.
11. A. AMIT and N. LINIAL: Random lifts of graphs: Edge expansion; Combinatorics Probability and Computing, to appear, 2005.
12. A. AMIT, N. LINIAL and J. MATOUŠEK: Random lifts of graphs: independence and chromatic number, Random Structures Algorithms 20(1) (2002), 1-22.
13. B. BOLLOBÁS and V. NIKIFOROV: Hermitian matrices and graphs: singular values and discrepancy; Discrete Math. 285(1-3) (2004), 17-32.
14. A. BRODER and E. SHAMIR: On the second eigenvalue of random regular graphs, in 28th Annual Symposium on Foundations of Computer Science, October 12-14, 1987, Los Angeles, California, pages 286-294; IEEE Comput. Soc. Press, 1987.
15. P. ERDOS and L. LOVÁSZ: Problems and results on 3-chromatic hypergraphs and some related questions, in Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdos on his 60th birthday), Vol. II, pages 609-627; Colloq. Math. Soc. János Bolyai, Vol. 10; North-Holland, Amsterdam, 1975.
16. J. FRIEDMAN: A proof of Alon's second eigenvalue conjecture, manuscript.
17. J. FRIEDMAN: On the second eigenvalue and random walks in random d-regular graphs, Combinatorica 11(4) (1991), 331-362.
18. J. FRIEDMAN: Relative expanders or weakly relatively Ramanujan graphs, Duke Math. J. 118(1) (2003), 19-35.
19. J. FRIEDMAN, J. KAHN and E. SZEMERÉDI: On the second eigenvalue of random regular graphs, in Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 587-598; ACM Press, 1989.
20. A. M. FRIEZE and M. MOLLOY: Splitting an expander graph, J. Algorithms 33(1) (1999), 166-172.
21. O. GABBER and Z. GALIL: Explicit constructions of linear-sized superconcentrators, J. Comput. System Sci. 22(3) (1981), 407-420 (Special issue dedicated to Michael Machtey).
22. R. A. HORN and C. R. JOHNSON: Matrix analysis, Cambridge University Press, Cambridge, 1990 (Corrected reprint of the 1985 original).
23. D. E. KNUTH: The art of computer programming, Vol.1: Fundamental algorithms; Second printing; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
24. N. LINIAL and E. ROZENMAN: Random lifts of graphs: Perfect matchings, Combinatorica 25(4) (2005), 407-424.
25. N. LINIAL and A. WIGDERSON: Expander graphs and their applications, Lecture notes available at http://www.math.ias.edu/~boaz/ ExpanderCourse/.
26. A. LUBOTZKY, R. PHILLIPS and P. SARNAK: Ramanujan graphs, Combinatorica 8(3) (1998), 261-277.
27. G. A. MARGULIS: Explicit constructions of expanders, Problemy Peredači Informacii 9(4) (1973), 71-80.
28. G. A. MARGULIS: Explicit group-theoretic constructio11ns of combinatorial schemes and their applications in the construction of expanders and concentrators, Problemy Peredachi Informatsii 24(1) (1988), 51-60.
29. J. NAOR and M. NAOR: Small-bias probability spaces: efficient constructions and applications, SIAM J. Comput. 22(4) (1993), 838-856.
30. A. NILLI: On the second eigenvalue of a graph, Discrete Math. 91(2) (1991), 207-210.
31. O. REINGOLD, S. VADHAN and A. WIGDERSON: Entropy waves, the zig-zag graph product, and new constant-degree expanders; Ann. of Math. (2) 155(1) (2002), 157-187.
32. A. THOMASON: Pseudorandom graphs, in Random graphs '85 (Poznań, 1985), volume 144 of North-Holland Math. Stud., pages 307-331; North-Holland, Amsterdam, 1987.