Maximal determinants of combinatorial matrices

TL;DR

This paper establishes upper bounds on the determinants of certain combinatorial matrices, including those with limited ones, the k-consecutive ones property, and path-edge incidence matrices, advancing understanding of their maximal determinants.

Contribution

It introduces new upper bounds for determinants of matrices with specific combinatorial structures, generalizing previous results and linking matrix determinants to graph properties.

Findings

Bound det A ≤ 6^{n/6} for matrices with at most 2n ones.

Derived bounds for matrices with the k-consecutive ones property.

Established an upper bound on the leaf rank of a graph based on its size.

Abstract

We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive ones property, where each row is allowed to have up to $k$ blocks of ones. Finally, we prove an upper bound on the determinant of a path-edge incidence matrix in a tree and use that to bound the leaf rank of a graph in terms of its order.