Maximal determinants of combinatorial matrices

We prove that det⁡A≤6n6 whenever A∈{0,1}n×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.