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.

Highlights

  • A fundamental result of Chung, Graham, and Wilson [6], building on earlier work of Thomason [19, 20], states that for a sequence of graphs of density p, where p is a fixed positive constant, a number of seemingly distinct notions of quasirandomness are equivalent

  • The adjacency matrix A(Γ) of an n-vertex graph Γ is the n × n matrixs,t∈[n] where ast = 1 if s and t are adjacent in Γ and 0 otherwise

  • We say that an n-vertex d-regular graph Γ = (V, E) is ε-uniform if, for all S, T ⊆ V, d e(S, T ) − |S||T | n

Read more

Summary

Introduction

A fundamental result of Chung, Graham, and Wilson [6], building on earlier work of Thomason [19, 20], states that for a sequence of graphs of density p, where p is a fixed positive constant, a number of seemingly distinct notions of quasirandomness are equivalent. The equivalence between properties P1 and P2 of Theorem 1.1 implies that if p is fixed and Γn is a sequence of graphs with |V (Γn)| = n and Γn regular of degree dn = pn, the sequence Γn is o(1)-uniform if and only if |λ2(Γn)| = o(dn) One direction of this equivalence follows from the famous expander mixing lemma. This says that if Γ = (V, E) is an (n, d, λ )-graph, that is, an n-vertex d-regular graph such that all eigenvalues of the adjacency matrix A(Γ), except the largest, are bounded above in absolute value by λ , d. A semidefinite relaxation was used by Alon et al [1] to prove the related result that if a graph G has small discrepancy, one can remove a small fraction of the vertices so that in the remaining graph all eigenvalues, except the largest, are small

Norms and relaxations
Cayley graphs
Transitive graphs
Bipartite analogues
A Representation-theoretic proof
Abelian groups
Group representation theory
Non-abelian Fourier analysis
Singular value decompositions
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call