Some graphs with small second eigenvalue


For any prime,p, we construct a Cayley graph on the group,G, of affine linear transformations ofℤ/pℤ of degree 2(p−1) and second eigenvalue\(2\sqrt p \) with the following special property: the adjacency matrix of the graph is supported on the “blocks” associated to the trivial representation and the irreducible representation of sizep−1. SinceG is of orderp(p−1), the correspondingt-uniform Cayley hypergraph has essentially optimal second eigenvalue for this degree and size of the graph (see [2] for definitions). En route we give, for any integerk>1, a simple Cayley graph onp k nodes of degreep of second eigenvalue\( \leqslant (k - 1)\sqrt p \).

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


  1. [1]

    F. R. K. Chung: Diameters and eigenvalues, preprint.

  2. [2]

    J. Friedman, andA. Wigderson: On the second eigenvalue of hypergraphs,Combinatorica 15 (1995), 43–65.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    H. Hasse: Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichen Konstantkörper.J. Reine Angew. Math.,172 (1934), 37–54.

    MATH  Google Scholar 

  4. [4]

    H. Hasse, andH. Davenport: Die nullstellen der kongruenzzetafunktionen in gewissen zyklischen fallen,J. Reine Angew. Math.,172 (1934), 151–182.

    MATH  Google Scholar 

  5. [5]

    Y. Ihara: Discrete subgroups of PL(2,k ), InProceedings of Symposia in Pure Mathematics, Volume IX: Algebraic Groups and Discontinuous Subgroups, 272–278, 1966.

  6. [6]

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

  7. [7]

    G. Margulis: Arithmetic groups and graphs without short cycles,6th Inter. Symp. on Info. Theory, Tashkent, Abstracts, Vol. I, 1984.

  8. [8]

    G. Margulis: Explicit group theoretic constructions of combinatorial schemes and their applications for construction of expanders and concentrators,J. of Problems of Information Transmission, 1988.

  9. [9]

    Alain Robert:Introduction to the Representation Theory of Compact and Locally Compact Groups, Cambridge University Press, Cambridge, 1983.

    Google Scholar 

  10. [10]

    Wolfgang M. Schmidt:Lecture Notes in Mathematics, #536: Equations over Finite Fields, An Elementary Approach, Springer-Verlag, New York, 1976.

    Google Scholar 

  11. [11]

    André Weil: On some exponential sums,Proc. Nat. Acad. Sci. USA,34 (1948), 204–207.

    MATH  MathSciNet  Article  Google Scholar 

  12. [12]

    André Weil:Basic Number Theory, Springer-Verlag, New York, 1974.

    Google Scholar 

Download references

Author information



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. Some graphs with small second eigenvalue. Combinatorica 15, 31–42 (1995).

Download citation

Mathematics Subject Classification (1991)

  • 05 C 50
  • 68 R 10
  • 05 C 65