On the girth of random cayley graphs

Random Structures and Algorithms

Volumn 35, Issue 1, 2009, Pages 100-117

  Gamburd, A ^a   Hoory, S ^b   Shahshahani, M ^c   Shalev, A ^d   Virag B ^e  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b IBM HAIFA RESEARCH LAB   (Israel)

^c INSTITUTE FOR RESEARCH IN FUNDAMENTAL SCIENCES IPM   (Iran)

^d HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^e UNIVERSITY OF TORONTO   (Canada)

Author keywords

Cayley graphs; Girth; Random

Indexed keywords

EID: 67650527011     PISSN: 10429832     EISSN: 10982418     Source Type: Journal    
DOI: 10.1002/rsa.20266     Document Type: Article

References (38)

1. M. Abért, Group laws and free subgroups in topological groups, Bull London Math Soc 37 (2005),525-534.
2. M. Abért, On the probability of satisfying a word in a group, J Group Theory, forthcoming math. GR/0504312. J Group Theory 9 (2006), 685-694.
3. M. Abért and B. Virag, Dimension and randomness in groups acting on rooted trees, J Amer Math Soc 18 (2005), 157-192.
4. N. Alon and Y. Roichman, Random Cayley graphs and expanders, Random Structures Algorithms 5 (1994), 271-284.
5. N. Alon and J. Spencer, The probabilistic method, Wiley, New York, 1992.
6. L. Babai, R. Beals, and A. Seress, On the diameter of the symmetric group: Polynomial bounds, Symposium on Discrete Algorithms, New Orleans, Louisiana, 2004, pp. 1108-1112.
7. L. Babai and T. P. Hayes, Near-independence of permutations and an almost-sure polynomial bound on the diameter of the symmetric group, SODA, Vancouver, British Columbia, 2005, pp. 1057-1066.
8. L. Babai and G. L. Hetyei, On the diameter of random Cayley graphs of the symmetric group, Combin Probab Comput 1 (1992), 201-208.
9. N. L. Biggs and A. G. Boshier, Note on the girth of Ramanujan graphs, J Combinatorial Theory B 49 (1990), 190-194.
10. N. L. Biggs and M. J. Hoare, The sextet construction for cubic graphs, Combinatorica 3 (1983), 153-165.
11. A. Borel, On free subgroups of algebraic groups, Enseign Math 29 (1983), 151-164.
12. p), Annals of Mathematics 167 (2008), 625-642.
13. J. Dixon, L. Pyber, A. Seress, and A. Shalev, Residual properties of free groups and probabilistic methods, J Reine Angew Math 556 (2003), 159-172.
14. D. B. A. Epstein, Almost all subgroups of a Lie group are free, J Algebra 19 (1971), 261-262.
15. P. Erdos and H. Sachs, Regulare Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss Z Univ Halle-Wittenberg Math-Nat R 12 (1963), 251-258.
16. A. Gamburd, D. Jakobson, and P. Sarnak, Spectra of elements in the group ring of SU(2), J Eur Math Soc 1 (1999), 51-85.
17. F. Hausdorff, Grundzuge der Mengenlehre, Leipzig, 1914.
18. E. Hrushovski, The first order theory of Frobenius automorphisms, (forthcoming).
19. W. Imrich, Explicit construction of graphs without small cycles, Combinatorica 2 (1984), 53-59.
20. S. Janson, T. Łuczak, and A. Rucinski, Random graphs, Wiley-Interscience, New York, 2000.
21. G. A. Jones, Varieties and simple groups, J Austr Math Soc 17 (1974), 163-173.
22. V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3), GAFA 11 (2001), 953-970.
23. A. Khinchine, Continued fractions (1963).
24. Z. Landau and A. Russell, Random Cayley graphs are expanders: A simple proof of the Alon-Roichman theorem, Electronic J Combinatorics 11, (2004), Research Paper 62.
25. M. Larsen, Word maps have large image, Israel J Math 139 (2004), 149-156.
26. F. Lazebnik, V. A. Ustimenko, and A. J. Woldar, A new series of dense graphs of high girth, Bull Amer Math Soc 32 (1995), 228-239.
27. M. W. Liebeck and A. Shalev, The probability of generating a finite simple group, Geometriae Dedicata 56 (1995), 103-113.
28. P. Loh and L. Schulman, Improved expansion of random Cayley graphs, Disc Math Theor Comp Sci 6 (2004), 523-528.
29. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan Graphs, Combinatorica 8 (1988), 261-277.
30. G. A. Margulis, Explicit Construction of Graphs without Short Cycles and Low Density Codes, Combinatorica 2 (1982), 71-78.
31. G. A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Prob Inform Trans 24 (1988), 39-46.
32. B. D. McKay, N. C. Wormald, and B. Wysocka, Short cycles in random regular graphs, The Electronic Journal of Combinatorics 11 (2004) #R66.
33. M. Morgenstern, Existence and explcit construction of q+1 regular Ramanujan graphs for every prime power q, J Combinatorial Theory B 62 (1994), 44-62.
34. I. Pak, Random Cayley graphs with O(log|G|) generators are expanders, Lecture Notes Comput Sci 1643 (1999), 521-526.
35. J. P. Serre, Lettre à M. Tsfasman, Astérisque 198-200 (1991), 351-353.
36. A.Shalev, Word maps conjugacy classes, andanon-commutative Waring-type theorem, Annals of Mathematics, forthcoming.
37. A. B. Sørensen, Projective Reed-Muller codes, IEEE Trans Inform Theory 37 (1991), 1567-1576.
38. A. I. Weiss, Girth of bipartite sexte graphs, Combinatorica 4 (1984), 241-245.