Quasirandom Cayley graphs

TL;DR

This paper establishes the equivalence of small discrepancy and small second eigenvalue in Cayley graphs, extending known results from abelian cases to all Cayley graphs and vertex-transitive graphs using Grothendieck's inequality.

Contribution

It generalizes a key spectral discrepancy equivalence from abelian to all Cayley graphs and vertex-transitive graphs, employing Grothendieck's inequality.

Findings

Equivalence of small discrepancy and small second eigenvalue in Cayley graphs.

Extension of the result to all vertex-transitive graphs.

Use of Grothendieck's inequality in the proof.

Abstract

We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, R\"odl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.