Maximum determinant and permanent of sparse 0-1 matrices

TL;DR

This paper establishes tight bounds on the maximum determinant of sparse 0-1 matrices and improves bounds on the number of perfect matchings in certain bipartite graphs, solving a longstanding conjecture.

Contribution

It proves a conjecture on the maximum determinant of sparse 0-1 matrices and provides improved bounds on perfect matchings in $C_4$-free bipartite graphs.

Findings

Maximum determinant bound of $2^{k/3}$ for sparse matrices.

Improved upper bound on perfect matchings in $C_4$-free bipartite graphs.

Tightness of bounds demonstrated by specific graph constructions.

Abstract

We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.