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Ramanujan graphs

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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 log k−1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.

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The work of the second author was supported in part by the NSF under the Grant No. DMS-85-03297 and the third by NSF Grant No. DMS-85-04329.

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Lubotzky, A., Phillips, R. & Sarnak, P. Ramanujan graphs. Combinatorica 8, 261–277 (1988).

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