Tight bounds for positive discrepancy via eigenvalues

Abstract

Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $\lambda_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $\lambda_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $\lambda$, then $\max_{v\in [-1,1]^n}v^TMv\geq \lambda(n-n'+1)$. We use this lemma to improve a number of recent results on the MaxCut, bisection width, and discrepancy of graphs. Among others, we prove that every $n$-vertex $m$-edge graph that is far from a disjoint union of cliques has a cut of size at least $m/2+n^{5/4-o(1)}$, which is sharp up to the $o(1)$-term. Moreover, we prove that every $d$-regular $n$-vertex graph has bisection width at most $dn/4-\Omega_{\varepsilon}(d^{1/3}n)$ for $d\leq (1-\varepsilon)n/2$, which is optimal for $d=\Omega(n)$. This confirms a conjecture of R\"aty, Sudakov and Tomon.