On laplacians of random complexes

Proceedings of the Annual Symposium on Computational Geometry

Volumn , Issue , 2012, Pages 151-160

  Gundert, Anna ^a   Wagner, Uli ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

Eigenvalues; Hypergraphs; Laplacians; Measure Concentration; Random Complexes; Simplicial Complexes

Indexed keywords

EIGENVALUES; HYPER GRAPH; LAPLACIANS; MEASURE CONCENTRATION; RANDOM COMPLEXES; SIMPLICIAL COMPLEX;

COMPUTATIONAL GEOMETRY; EXPANSION; GRAPH THEORY; LAPLACE TRANSFORMS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84863930007     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/2261250.2261272     Document Type: Conference Paper

References (64)

1. R. M. Adin. Counting colorful multi-dimensional trees. Combinatorica, 12(3):247-260, 1992.
2. R. Aharoni, E. Berger, and R. Meshulam. Eigenvalues and homology of flag complexes and vector representations of graphs. Geom. Funct. Anal., 15(3):555-566, 2005.
3. N. Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986.
4. 1, Isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1):73-88, 1985.
5. N. Alon, O. Schwartz, and A. Shapira. An elementary construction of constant-degree expanders. Combin. Probab. Comput., 17(3):319-327, 2008.
6. L. Aronshtam, N. Linial, T. Luczak, and R. Meshulam. Collapsibility and vanishing of top homology in random simplicial complexes. Preprint, arXiv:1010.1400v2, 2010.
7. E. Babson, C. Hoffman, and M. Kahle. The fundamental group of random 2-complexes. J. Amer. Math. Soc., 24(1):1-28, 2011.
8. A. Borel. Cohomologie des certains groupes discrets et Laplacien p-adique. In Séminaire Bourbaki, Exposé 437, volume 431 of Lecture Notes in Math., pages 12-35. Springer, 1973.
9. A. Cayley. A theorem on trees. Quart. J. Pure Appl. Math., 23:376-378, 1889. Collected Mathematical Papers Vol. 13, Cambridge Univ. Press, 1897, 26-28.
10. J. Cheeger. A lower bound for the smallest eigenvalue of the Laplacian. In Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pages 195-199. Princeton Univ. Press, Princeton, N. J., 1970.
11. F. Chung. The Laplacian of a Hypergraph, volume 10 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 21-36. Amer. Math. Soc., 1993.
12. F. Chung. Spectral graph theory, volume 92 of CBMS Reg. Conf. Ser. Math. Amer. Math. Soc., 1997.
13. F. Chung, L. Lu, and V. Vu. The spectra of random graphs with given expected degrees. Internet Math., 1(3):257-275, 2004.
14. D. C. Cohen, M. Farber, and T. Kappeler. The homotopical dimension of random 2-complexes. Preprint, arXiv:1005.3383v1, 2010.
15. n,p. Combin. Probab. Comput., 16(6):923-946, 2007.
16. A. Coja-Oghlan and A. Lanka. The spectral gap of random graphs with given expected degrees. Electron. J. Combin., 16(1):R138, 2009.
17. A. Costa, M. Farber, and T. Kappeler. Topology of random 2-complexes. Preprint, arXiv:1006.4229v2, 2010.
18. D. Cvetković, P. Rowlinson, and S. Simić. An introduction to the theory of graph spectra, volume 75 of London Math. Soc. Stud. Texts. Cambridge Univ. Press, Cambridge, 2010.
19. J. Dodziuk. Finite-difference approach to the Hodge theory of harmonic forms. Amer. J. Math., 98(1):79-104, 1976.
20. J. Dodziuk. Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc., 284(2):787-794, 1984.
21. J. Dodziuk and V. K. Patodi. Riemannian structures and triangulations of manifolds. J. Indian Math. Soc. (N.S.), 40(1-4):1-52 (1977), 1976.
22. X. Dong and M. L. Wachs. Combinatorial Laplacian of the matching complex. Electron. J. Combin., 9(1):Research Paper 17, 11, 2002.
23. D. Dotterrer and M. Kahle. Coboundary expanders. Preprint, arXiv:1012.5316, 2010.
24. A. M. Duval, C. J. Klivans, and J. L. Martin. Simplicial matrix-tree theorems. Trans. Amer. Math. Soc., 361(11):6073-6114, 2009.
25. B. Eckmann. Harmonische Funktionen und Randwertaufgaben in einem Komplex. Comment. Math. Helv., 17:240-255, 1945.
26. U. Feige and E. Ofek. Spectral techniques applied to sparse random graphs. Random Structures Algorithms, 27(2):251-275, 2005.
27. J. Fox, M. Gromov, V. Lafforgue, A. Naor, and J. Pach. Overlap properties of geometric expanders. Preprint, arXiv:1005.1392, 2010.
28. J. Friedman. Some geometric aspects of graphs and their eigenfunctions. Duke Math. J., 69(3):487-525, 1993.
29. J. Friedman. Computing Betti numbers via combinatorial Laplacians. Algorithmica, 21(4):331-346, 1998.
30. J. Friedman and P. Hanlon. On the Betti numbers of chessboard complexes. J. Algebraic Combin., 8(2):193-203, 1998.
31. J. Friedman, J. Kahn, and E. Szemerédi. On the second eigenvalue of random regular graphs. In Proc. 21st Ann. ACM Symp. Theor. Comput., STOC '89, pages 587-598, New York, 1989. ACM.
32. Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica, 1(3):233-241, 1981.
33. O. Gabber and Z. Galil. Explicit constructions of linear-sized superconcentrators. J. Comput. System Sci., 22(3):407-420, 1981.
34. H. Garland. p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Ann. of Math. (2), 97(3):375-423, 1973.
35. W. T. Gowers. Quasirandomness, counting and regularity for 3-uniform hypergraphs. Combin. Probab. Comput., 15(1-2):143-184, 2006.
36. M. Gromov. Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal., 20(2):416-526, 2010.
37. W. V. D. Hodge. The Theory and Applications of Harmonic Integrals. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge, 1989.
38. C. Hoffman, M. Kahle, and E. Paquette. A sharp threshold for Kazhdan's property (T). Preprint, arXiv:1201.0425, 2012.
39. S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4):439-561 (electronic), 2006.
40. D. Horak and J. Jost. Spectra of combinatorial Laplace operators on simplicial complexes. Preprint, arXiv:1105.2712v3, 2011.
41. S. Janson. On concentration of probability. In B. Bollobás, editor, Contemporary Combinatorics, volume 10 of Boly. Soc. Math. Stud., pages 289-301. Springer, 2002.
42. S. Janson, T. Luczak, and A. Ruciński. Random Graphs. Wiley-Intersci. Ser. in Discrete Math. Optim. Wiley, 2000.
43. M. Jerrum. Counting, sampling and integrating: algorithms and complexity. Lectures Math. ETH Zürich. Birkhäuser, Basel, 2003.
44. G. Kalai. Enumeration of Q-acyclic simplicial complexes. Israel J. Math., 45:337-351, 1983.
45. R. N. Karasev. A Simpler Proof of the Boros-Füredi-Barany-Pach- Gromov Theorem. Discrete Comput. Geom., 47(3):492-495, 2012.
46. G. Kirchhoff. Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys., 148(12):497-508, 1847.
47. W. Kook, V. Reiner, and D. Stanton. Combinatorial Laplacians of matroid complexes. J. Amer. Math. Soc, 13(1):129-148, 2000.
48. D. Kozlov. The threshold function for vanishing of the top homology group of random d-complexes. Proc. Amer. Math. Soc, 138(12):4517-4527, 2010.
49. M. Krivelevich and B. Sudakov. Pseudo-random graphs. In More sets, graphs and numbers, volume 15 of Bolyai Soc. Math. Stud., pages 199-262. Springer, 2006.
50. D. A. Levin, Y. Peres, and E. L. Wilmer. Markov chains and mixing times. Amer. Math. Soc, 2009.
51. N. Linial and R. Meshulam. Homological connectivity of random 2-complexes. Combinatorica, 26(4):475-487, 2006.
52. L. Lu and X. Peng. Loose Laplacian spectra of random hypergraphs. Preprint, arXiv:1109.3433, 2011.
53. A. Lubotzky, R. S. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988.
54. G. A. Margulis. Explicit constructions of expanders. Problemy Peredaci Informacii, 9(4):71-80, 1973.
55. G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1):51-60, 1988.
56. J. Matousek and U. Wagner. On Gromov's method of selecting heavily covered points. Preprint, arXiv:1102.3515v1, 2011.
57. R. Meshulam and N. Wallach. Homological connectivity of random k-dimensional complexes. Random Structures Algorithms, 34(3):408-417, 2009.
58. I. Newman and Y. Rabinovich. On multiplicative A-approximations and some geometric applications. In Proc. 23rd Ann. ACM-SIAM Symp. Discrete Algorithms. SIAM, 2011.
59. A. Nilli. On the second eigenvalue of a graph. Discrete Math., 91(2):207-210, 1991.
60. R. I. Oliveira. Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. Preprint, arXiv:0911.0600v2, 2010.
61. 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): 157-187, 2002.
62. U. Wagner. Minors in random and expanding hypergraphs. In Proc. 27th Ann. Symp. Comput. Geom. (SoCG), pages 351-360, 2011.
63. X.-D. Zhang. The Laplacian eigenvalues of graphs: a survey. Preprint, arXiv:1111.2897v1, 2011.
64. A. Zuk. La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres. C. R. Acad. Sci. Paris, 323:453-458, 1996.