Discrepancy and Eigenvalues of Cayley Graphs
arXiv:1602.02291 · doi:10.1007/s10587-016-0302-x
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 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.
Dedicated to the memory of Professor Miroslav Fiedler, 33 pages, second version addresses changes arising from the referee report and includes an appendix with an earlier argument