Bisection Width, Discrepancy, and Eigenvalues of Hypergraphs

A celebrated result of Alon from 1993 states that any $d$-regular graph on $n$ vertices (where $d=O(n^{1/9})$) has a bisection with at most $\frac{dn}{2}(\frac{1}{2}-\Omega(\frac{1}{\sqrt{d}}))$ edges, and this is optimal. Recently, this result was greatly extended by R\"aty, Sudakov, and Tomon. We build on the ideas of the latter, and use a semidefinite programming inspired approach to prove the following variant for hypergraphs: every $r$-uniform $d$-regular hypergraph on $n$ vertices (where $d\ll n^{1/2}$) has a bisection of size at most $$\frac{dn}{r}\left(1-\frac{1}{2^{r-1}}-\frac{c}{\sqrt{d}}\right),$$ for some $c=c(r)>0$. This bound is the best possible up to the precise value of $c$. Moreover, a bisection achieving this bound can be found by a polynomial-time randomized algorithm. The minimum bisection is closely related to discrepancy. We also prove sharp bounds on the discrepancy and so called positive discrepancy of hypergraphs, extending results of Bollob\'as and Scott. Furthermore, we discuss implications about Alon-Boppana type bounds. We show that if $H$ is an $r$-uniform $d$-regular hypergraph, then certain notions of second largest eigenvalue $\lambda_2$ associated with the adjacency tensor satisfy $\lambda_2\geq \Omega_r(\sqrt{d})$, improving results of Li and Mohar.