Abstract
We prove that the maximum determinant of an n×n matrix, with entries in {0,1} and at most n+k non-zero entries, is at most 2k/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 C4-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.
Original language | English (US) |
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Pages (from-to) | 194-228 |
Number of pages | 35 |
Journal | Linear Algebra and Its Applications |
Volume | 645 |
DOIs | |
State | Published - Jul 15 2022 |
Keywords
- 0,1 matrices
- Determinant
- Permanent
- Sparse matrices
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics