On the second eigenvalue and random walks in randomd-regular graphs

Abstract

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, λ2, of (the adjacency matrix of) a randomd-regular graph,G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at thek-th step, for various values ofk. Our main theorem about eigenvalues is that

$$E|\lambda _2 (G)|^m \leqslant \left( {2\sqrt {2d - 1} \left( {1 + \frac{{\log d}}{{\sqrt {2d} }} + 0\left( {\frac{1}{{\sqrt d }}} \right)} \right) + 0\left( {\frac{{d^{3/2} \log \log n}}{{\log n}}} \right)} \right)^m $$

for any\(m \leqslant 2\left\lfloor {log n\left\lfloor {\sqrt {2d - } 1/2} \right\rfloor /\log d} \right\rfloor \), where E denotes the expected value over a certain probability space of 2d-regular graphs. It follows, for example, that for fixedd the second eigenvalue's magnitude is no more than\(2\sqrt {2d - 1} + 2\log d + C'\) with probability 1−n −C for constantsC andC′ for sufficiently largen.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon: Eigenvalues and expanders.Combinatorica 6 (2):83–96, 1986.

    Google Scholar 

  2. [2]

    Andrei Broder, andEli Shamir: On the second eigenvalue of random regular graphs. In28th annual Symposium on Foundations of Computer Science, pages 286–294, 1987.

  3. [3]

    Z. Füredi, andJ. Komlós: The eigenvalues of random symmetric matrices.Combinatorica,1 (3): 233–241, 1981.

    Google Scholar 

  4. [4]

    S. Geman: A limit theorem for the norm of random matrices.Ann. of Prob.,8 (2): 252–261, 1980.

    Google Scholar 

  5. [5]

    Donald E. Knuth:The Art of Computer Programming, volume 1: Fundamental Algorithms. Addison-Wesley, Reading, Massachusetts, second edition, 1973.

    Google Scholar 

  6. [6]

    J. Kahn, andE. Szemerédi: Personal communication, and to appear.

  7. [7]

    A. Lubotzky, R. Phillips, andP. Sarnak: Explicit expanders and the Ramanujan conjectures. In18th Annual ACM Symposium on Theory of Computing, pages 240–246, 1986.

  8. [8]

    G. Margulis: Manuscript in Russian on graphs with large girth, 1987.

  9. [9]

    B. McKay: The expected eigenvalue distribution of a large regular graph.Lin. Alg. Appl.,40: 203–216, 1981.

    Google Scholar 

  10. [10]

    R. M. Tanner: Explicit concentrators from generalizedN-gons.SIAM J. Alg. Disc. Methods,5: 287–293, 1984.

    Google Scholar 

  11. [11]

    E. Wigner: Characteristic vectors of bordered matrices with infinite dimensions.Annals of Math. 62 (3): 548–564, 1955.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and the Office of Naval Research under Grant N00014-87-K-0467.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Friedman, J. On the second eigenvalue and random walks in randomd-regular graphs. Combinatorica 11, 331–362 (1991). https://doi.org/10.1007/BF01275669

Download citation

AMS subject classification code (1991)

  • 05 C 50
  • 05 C 80