Discrepancy and eigenvalues of Cayley graphs

Yoshiharu Kohayakawa, Vojtěch Rödl, Mathias Schacht
2016 Czechoslovak Mathematical Journal  
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 positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.
doi:10.1007/s10587-016-0302-x fatcat:tw7ohim7hbcj5a2tborfaijrmu