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Eigenvalues and expanders

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Abstract

Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expanderif and only if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties. It also supplies an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete.

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References

  1. H. Abelson, A note on time space tradeoffs for computing continuous functions,Infor. Proc. Letters 8 (1979), 215–217.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Ajtai, J. Komlós andE. Szemerédi, Sorting inc logn parallel steps,Combinatorica 3 (1983), 1–19.

    MATH  MathSciNet  Google Scholar 

  3. N. Alon andV. D. Milman, λ1, isoperimetric inequalities for graphs and superconcentrators,J. Combinatorial Theory Ser. B,38 (1985), 73–88.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. Alon andV. D. Milman, Eigenvalues, expanders and superconcentrators,Proc. 25 th Ann. Symp. on Foundations of Comp. Sci., Florida (1984), 320–322.

  5. N. Alon, Z. Galil andV. D. Milman, Better expanders and superconcentrators,to appear.

  6. W. N. Anderson, Jr. andT. D. Morley, Eigenvalues of the Laplacian of a graph,University of Maryland Technical Report TR-71-45, (1971).

  7. L. A. Bassalygo, Asymptotically optimal switching circuits,Problems of Infor. Trans. 17 (1981), 206–211.

    MATH  Google Scholar 

  8. N. Biggs,Algebraic Graph Theory, Cambridge University Press, London, 1974.

    MATH  Google Scholar 

  9. M. Blum, R. M. Karp, O. Vornberger, C. H. Papadimitriou andM. Yannakakis, The complexity of testing whether a graph is a superconcentrator,Inform. Process. Letters 13 (1981), 164–167.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Bollobás, A. probabilistic proof of an asymptotic formula for the number of labelled regular graphs,Europ. J. Combinatorics 1 (1980), 311–316.

    MATH  Google Scholar 

  11. P. Buser, Cubic graphs and the first eigenvalue of a Riemann surface,Math. Z. 162 (1978), 87–99.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, inProblems in Analysis (edited by R. C. Gunning), Princeton University Press, New Jersey, (1970), 195–199.

    Google Scholar 

  13. F. K. R. Chung, On concentrators, superconcentrators, generalizers and nonblocking networks,Bell Sys. Tech. J. 58 (1978), 1765–1777.

    Google Scholar 

  14. D. M. Cvetkovic, Some possible directions in further investigations of graph spectra, in:Algebraic Methods in Graph Theory (Coll. Math. Soc. J. Bolyai, L. Lovász and V. T. Sós eds.), North-Holland, Amsterdam, (1981) 47–67.

    Google Scholar 

  15. M. Fiedler, Algebraic connectivity of graphs,Czechoslovak. Math. J. 23 (98), (1973), 298–305.

  16. Z. Füredi andJ. Komlós, The eigenvalues of random symmetric matrices,Combinatorica 1 (1981), 233–241.

    MATH  MathSciNet  Google Scholar 

  17. O. Gabber andZ. Galil, Explicit construction of linear sized superconcentrators,J. Comp. and Sys. Sci. 22 (1981), 407–420.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Gromov andV. D. Milman, A topological application of the isoperimetric inequality,American J. Math. 105 (1983), 843–854.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Ja’Ja, Time space tradeoffs for some algebraic problems,Proc. 12 th Ann. ACM Symp. on Theory of Computing, (1980), 339–350.

  20. S. Jimbo andA. Maruoka, Expanders obtained from affine transformations (extended abstract),preprint (1984).

  21. T. Lengauer andR. E. Tarjan, Asymptotically tight bounds on time space tradeoffs in a pebble game,J. ACM 29 (1982), 1087–1130.

    Article  MATH  MathSciNet  Google Scholar 

  22. G. A. Margulis, Explicit constructions of concentrators,Prob. Per. Infor. 9 (1973), 71–80. (English translation inProblems of Infor. Trans. (1975), 325–332.

    MathSciNet  MATH  Google Scholar 

  23. B. D. McKay, The expected eigenvalue distribution of a large regular graph,Linear Algebra Appl. 40 (1981), 203–216.

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Pinkser, On the complexity of a concentrator,7 th International Teletraffic Conference, Stockholm, June 1973, 318/1–318/4.

  25. N. Pippenger, Superconcentrators,SIAM J. Computing 6 (1977), 298–304.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. Pippenger, Advances in pebbling,Internat. Colloq. on Autom., Langs. and Prog. 9 (1982), 407–417.

    Article  Google Scholar 

  27. W. J. Paul, R. E. Tarjan andJ. R. Celoni, Space bounds for a game on graphs,Math. Sys. Theory 10 (1977), 239–251.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Ralston,A First Course in Numerical Analysis, McGraw-Hill, 1985, Section 10.4.

  29. A. L. Rosenberg, Fault tolerant interconnection networks; a graph theoretic approach, in:Proc. of a Workshop on Graph Theoretic Concepts in Computer Science, Trauner Verlag, Linz (1983), 286–297.

    Google Scholar 

  30. R. M. Tanner, Explicit construction of concentrators from generalizedN-gons,SIAM J. Alg. Discr. Meth. 5 (1984), 287–293.

    Article  MATH  MathSciNet  Google Scholar 

  31. M. Tompa, Time space tradeoffs for computing functions using connectivity properties of their circuits,J. Comp. and Sys. Sci. 20 (1980), 118–132.

    Article  MATH  MathSciNet  Google Scholar 

  32. L. G. Valiant, Graph theoretic properties in computational complexity,J. Comp. and Sys. Sci. 13 (1976), 278–285.

    MATH  MathSciNet  Google Scholar 

  33. E. P. Wigner, On the distribution of the roots of certain symmetric matrices,Ann. Math. 67 (1958), 325–327.

    Article  MathSciNet  Google Scholar 

  34. A. Lubotzky, R. Phillips, andP. Sarnak, Ramanujan graphs,to appear.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Inst. of Technology, 02139, Cambridge, MA, USA

    Noga Alon

  2. Department of Mathematics, Tel Aviv University, 69978, Ramat Aviv, Tel Aviv, Israel

    Noga Alon

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  1. Noga Alon
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Additional information

The research was supported by the Weizmann Fellowship for Scientific Research and by Air Force Contract OSR 82-0326.

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Alon, N. Eigenvalues and expanders. Combinatorica 6, 83–96 (1986). https://doi.org/10.1007/BF02579166

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  • DOI: https://doi.org/10.1007/BF02579166

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