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On the Laplacian Eigenvalues of Gn,p


We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of is 1-O (d−1/2) w.h.p.

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[1]Alon, N. (1998) Spectral techniques in graph algorithms. In Proc. 3rd LATIN, pp. 206215.
[2]Alon, N. and Kahale, N. (1997) A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26 17331748.
[3]Alon, N., Krivelevich, M. and Sudakov, B. (1998) Finding a large hidden clique in a random graph. Random Struct. Alg. 13 457466.
[4]Bauer, M. and Golinelli, O. (2001) Random incidence matrices: Moments of the spectral density. J. Statist. Phys. 103 301337.
[5]Bollobás, B. (2001) Random Graphs, 2nd edn, Springer.
[6]Boppana, R. (1987) Eigenvalues and graph bisection: An average-case analysis. In Proc. 28th FOCS, pp. 280285.
[7]Chung, F. R. K. (1997) Spectral Graph Theory, AMS.
[8]Chung, F. R. K., Lu, L. and Vu, V. (2003) The spectra of random graphs with given expected degrees. Internet Mathematics 1 257275.
[9]Coja-Oghlan, A. (2005) Spectral techniques, semidefinite programs, and random graphs. Habilitation thesis, Humboldt University, Berlin.
[10]Coja-Oghlan, A. (2006) A spectral heuristic for bisecting random graphs. Random Struct. Alg. 29 351398.
[11]Dasgupta, A., Hopcroft, J. E. and McSherry, F. (2004) Spectral partitioning of random graphs. In Proc. 45th FOCS, pp. 529537.
[12]Donath, W. E. and Hoffman, A. J. (1973) Lower bounds for the partitioning of a graph. IBM J. Res. Develop. 17 420425.
[13]Feige, U. and Ofek, E. (2005) Spectral techniques applied to sparse random graphs. Random Struct. Alg. 27 251275.
[14]Friedman, J. (2003) A proof of Alon's second eigenvalue conjecture. In Proc. 35th STOC, pp. 720724.
[15]Friedman, J., Kahn, J. and Szemerédi, E. (1989) On the second eigenvalue in random regular graphs. In Proc. 21st STOC, pp. 587598.
[16]Füredi, Z. and Komloś, J. (1981) The eigenvalues of random symmetric matrices. Combinatorica 1 233241.
[17]Golub, G. H. and van Loan, C. F. (1996) Matrix Computations, 3rd edn, Johns Hopkins University Press.
[18]Guattery, S. and Miller, G. L. (1998) On the quality of spectral separators. SIAM J. Matrix Anal. Appl. 19 701719.
[19]Janson, S. (2005) The first eigenvalue of random graphs. Combin. Probab. Comput. 14 815828.
[20]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.
[21]Khorunzhiy, O., Kirsch, W. and Müller, P. (2005) Lifshits tails for spectra of Erdos–Renyi random graphs. Annals of applied probability 16 (2006) 295309. Preprint: arXiv:math-ph/0502054 v1.
[22]Krivelevich, M. and Sudakov, B. (2003) The largest eigenvalue of sparse random graphs. Combin. Probab. Comput. 12 6172.
[23]Pothen, A., Simon, H. D. and Liou, K.-P. (1990) Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11 430452.
[24]Schloegel, K., Karypis, G. and Kumar, V. (2000) Graph partitioning for high performance scientific simulations. In CRPC Parallel Computation Handbook (Dongarra, J., Foster, I., Fox, G., Kennedy, K. and White, A., eds), Morgan-Kaufmann.
[25]Vu, V. (2005) Spectral norm of random matrices. In Proc. 37th STOC, pp. 423430.
[26]Wigner, E. P. (1958) On the distribution of the roots of certain symmetric matrices. Ann. of Math. 67 325327.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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