## Abstract

We study the Boolean rank of two families of binary matrices. The first is the binary matrix A _{k,t} that represents the adjacency matrix of the intersection bipartite graph of all subsets of size t of {1,2,…,k}. We prove that its Boolean rank is k for every k≥2t. The second family is the family U _{s,m} of submatrices of A _{k,t} that is defined as U _{s,m} =(J _{m} ⊗I _{s} )+(I¯ _{m} ⊗J _{s} ), where I _{s} is the identity matrix, J _{s} is the all-ones matrix, s=k−2t+2 and m=(2t−2t−1). We prove that the Boolean rank of U _{s,m} is also k for the following values of t and s: for s=2 and any t≥2, that is k=2t; for t=3 and any s≥2; and for any t≥2 and s>2t−2, that is k>4t−4.

Original language | English |
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Pages (from-to) | 67-83 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 574 |

DOIs | |

State | Published - 1 Aug 2019 |

## Keywords

- Boolean rank
- Cover size
- Intersection matrix