, Volume 8, Issue 3, pp 261–277 | Cite as

Ramanujan graphs

  • A. Lubotzky
  • R. Phillips
  • P. Sarnak


A large family of explicitk-regular Cayley graphsX is presented. These graphs satisfy a number of extremal combinatorial properties.
  1. (i)

    For eigenvaluesλ ofX eitherλ=±k or ¦λ¦≦2 √k−1. This property is optimal and leads to the best known explicit expander graphs.

  2. (ii)

    The girth ofX is asymptotically ≧4/3 logk−1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.


AMS subject classification (1980)



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Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • A. Lubotzky
    • 1
  • R. Phillips
    • 2
  • P. Sarnak
    • 3
  1. 1.Institute of Mathematics and Computer ScienceHebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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