Hermitian matrices and graphs: Singular values and discrepancy

Abstract

Let A=(aij)i,j=1n be a Hermitian matrix of size n≥2, and set ρ(A)=1n2∑i,j=1naij,disc(A)= maxX,Y⊂[n],X≠Θ,Y≠Θ1|X||Y|∑i∈X∑j∈Y(a ij-ρ(A)).We show that the second singular value σ2(A) of A satisfiesσ2(A)≤C 1disc(A)lognfor some absolute constant C1, and this is best possible up to a multiplicative constant. Moreover, we construct infinitely many dense regular graphs G such thatσ2(A(G))≥C 2disc(A(G))log|G|,where C20 is an absolute constant and A(G) is the adjacency matrix of G. In particular, these graphs disprove two conjectures of Fan Chung. © 2004 Elsevier B.V. All rights reserved.