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Discrepancy and eigenvalues of Cayley graphs

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Abstract

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, R. do Matão, 1010–Vila Universitaria, 05508-090, São Paulo, Brazil

    Yoshiharu Kohayakawa

  2. Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., Atlanta, Georgia, 30307, USA

    Vojtěch Rödl

  3. Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Mathias Schacht

Authors
  1. Yoshiharu Kohayakawa
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  2. Vojtěch Rödl
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  3. Mathias Schacht
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Correspondence to Yoshiharu Kohayakawa.

Additional information

Dedicated to the memory of Professor Miroslav Fiedler

The first author was supported by FAPESP (2013/03447-6, 2013/07699-0), CNPq (459335/2014-6, 310974/2013-5) and Project MaCLinC/USP. The second author was supported by NSF grant DMS 1301698. The third author was supported through the Heisenberg-Programme of the DFG. The collaboration of the first and third authors is supported by CAPES/DAAD PROBRAL project 430/15.

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Kohayakawa, Y., Rödl, V. & Schacht, M. Discrepancy and eigenvalues of Cayley graphs. Czech Math J 66, 941–954 (2016). https://doi.org/10.1007/s10587-016-0302-x

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