Discrepancy and Eigenvalues of Cayley Graphs

TL;DR

This paper establishes the equivalence of uniform edge-distribution and large eigenvalue gap in sparse Cayley graphs of finite abelian groups, advancing understanding of their quasirandom properties.

Contribution

It proves that for Cayley graphs of finite abelian groups, small discrepancy and large eigenvalue gap are equivalent, answering a specific open question.

Findings

Uniform edge-distribution implies large eigenvalue gap in these graphs

Large eigenvalue gap implies uniform edge-distribution in these graphs

The equivalence holds even for sparse Cayley graphs of abelian groups

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.