Positive discrepancy, MaxCut, and eigenvalues of graphs

TL;DR

This paper investigates the positive discrepancy of graphs, providing new bounds based on average degree and extending classical results, with implications for eigenvalues and graph cuts.

Contribution

It extends Alon's positive discrepancy bounds to broader degree ranges and establishes new lower bounds on eigenvalues for regular graphs using semidefinite programming.

Findings

Discrepancy bounds vary with degree ranges, optimal for certain regimes.

Regular graphs with specific degree ratios have large cuts, exceeding random expectations.

Lower bounds on second eigenvalues extend Alon-Boppana theorem in dense graphs.

Abstract

The positive discrepancy of a graph $G$ of edge density $p=e(G)/\binom{v(G)}{2}$ is defined as $$\mbox{disc}^{+}(G)=\max_{U\subset V(G)}e(G[U])-p\binom{|U|}{2}.$$ In 1993, Alon proved (using the equivalent terminology of minimum bisections) that if $G$ is $d$-regular on $n$ vertices, and $d=O(n^{1/9})$, then $\mbox{disc}^{+}(G)=\Omega(d^{1/2}n)$. We greatly extend this by showing that if $G$ has average degree $d$, then $\mbox{disc}^{+}(G)=\Omega(d^{\frac{1}{2}}n)$ if $d\in [0,n^{\frac{2}{3}}]$, $\Omega(n^2/d)$ if $d\in [n^{\frac{2}{3}},n^{\frac{4}{5}}]$, and $\Omega(d^{\frac{1}{4}}n/\log n)$ if $d\in \left[n^{\frac{4}{5}},(\frac{1}{2}-\varepsilon)n\right]$. These bounds are best possible if $d\ll n^{3/4}$, and the complete bipartite graph shows that $\mbox{disc}^{+}(G)=\Omega(n)$ cannot be improved if $d\approx n/2$. Our proofs are based on semidefinite programming and linear algebraic techniques. An interesting corollary of our results is that every $d$-regular graph on $n$ vertices with ${\frac{1}{2}+\varepsilon\leq \frac{d}{n}\leq 1-\varepsilon}$ has a cut of size $\frac{nd}{4}+\Omega(n^{5/4}/\log n)$. This is not necessarily true without the assumption of regularity, or the bounds on $d$. The positive discrepancy of regular graphs is controlled by the second eigenvalue $\lambda_2$, as $\mbox{disc}^{+}(G)\leq \frac{\lambda_2}{2} n+d$. As a byproduct of our arguments, we present lower bounds on $\lambda_2$ for regular graphs, extending the celebrated Alon-Boppana theorem in the dense regime.