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Eigenvalues of Graphs and Sobolev Inequalities

Published online by Cambridge University Press:  12 September 2008

F. R. K. Chung  and
S.-T. Yau
F. R. K. Chung
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 E-mail: [email protected]
S.-T. Yau
Affiliation:
Harvard University, Cambridge, Massachusetts 02138
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Abstract

We derive bounds for eigenvalues of the Laplacian of graphs using discrete versions of the Sobolev inequalities and heat kernel estimates.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 1 , March 1995 , pp. 11 - 25
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Alon, N. and Milman, V. D. (1985) λ1 isoperimetric inequalities for graphs and super-concentrators. J. Comb. Theory B 38, 73–88.CrossRefGoogle Scholar
[2]Babai, L. and Szegedy, M. (1992) Local expansion of symmetrical graphs. Combinatorics, Probability and Computing 1, 1–12.CrossRefGoogle Scholar
[3]Bollobás, B. (1978) Extremal Graph Theory, Academic Press, London.Google Scholar
[4]Brooks, R. (to appear) The spectral geometry of k-regular graphs. J. d'analyse.Google Scholar
[5]Buser, P. (1988) Cayley graphs and planar isospectral domains. In: Sunada, T. (ed.) Geometry and Analysis on Manifords, Springer Lecture Notes 1339, 64–77.Google Scholar
[6]Cheeger, J. (1970) A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R. C. (ed.) Problems in Analysis, Princeton University Press, 195–199.Google Scholar
[7]Chavel, I. (1984) Eigenvalues in Riemannian Geometry, Academic Press.Google Scholar
[8]Chung, F. R. K. (1995) Spectral Graph Theory, Lecture Notes of CBMS Regional Conference Series in Mathematics.CrossRefGoogle Scholar
[9]Chung, F. R. K. (preprint). Eigenvalues of graphs and Cheeger inequalities.Google Scholar
[10]Chung, F. R. K. and Yau, S.-T. (to appear) Haraack inequalities for graphs and subgraphs. Comm. on Analysis and Geometry.Google Scholar
[11]Chung, F. R. K., Grigor'yan, A. and Yau, S.-T. (to appear). Eigenvalue upper bounds of the Laplace operator on manifolds and graphs. Advances.Google Scholar
[12]Diaconis, P. and Saloff-Coste, L. (preprint). An application of Harnack inequalities to random walk on nilpotent quotients.Google Scholar
[13]Dodziuk, J. and Karp, L. (1988) Spectral and function theory for combinatorial Laplacians, Geometry of Random Motion. Contemp. Math 73, 25–40.CrossRefGoogle Scholar
[14]Gromov, M. (1981) Groups of polynomial growth and expanding maps. Publ. IHES. 53, 53–78.CrossRefGoogle Scholar
[15]Hebisch, W. and Saloff-Coste, L. (preprint). Gaussian estimates for Markov chains and random walks on groups.Google Scholar
[16]Li, P. and Yau, S.-T. (1980) Estimates of eigenvalues of a compact Riemannian manifold. Amer. Math. Soc. Proc. Symp. Pure Math. 36, 205–240.CrossRefGoogle Scholar
[17]Lubotsky, A., Phillips, R. and Sarnak, P. (1988) Ramanujan graphs. Combinatorica 8, 261–278.CrossRefGoogle Scholar
[18]Margulis, G. A. (1988) Explicit group theoretic constructions of combinatorial schemes and their applications for the construction of expanders and concentrators. Journal of Problems of Information Transmission (in Russian).Google Scholar
[19]Mohar, B. (1989) Isoperimetric number of graphs. J. of Comb. Theory (B) 47, 274–291.CrossRefGoogle Scholar
[20]Polya, G. (1961) On the eigenvalues of vibrating membranes. Proc. London Math. Soc. 11, 419–433.CrossRefGoogle Scholar
[21]Sarnak, P. (1990) Some Applications of Modular Forms. Cambridge University Press.CrossRefGoogle Scholar
[22]Sinclair, A. J. and Jerrum, M. R. (to appear). Approximate counting, uniform generation, and rapidly mixing Markov chains. Information and Computation.Google Scholar
[23]Varopoulos, N. Th. (1985) Isoperimetric inequalities and Markov chains. J. Fund. Anal. 63, 215–239.CrossRefGoogle Scholar
[24]Yau, S.-T. and Schoen, R. M. (1988). Differential Geometry, Science Publication Co. (in Chinese).Google Scholar