#### Title

Boolean Rank of Upset Tournament Matrices

#### Document Type

Article

#### Journal/Book Title/Conference

Linear Algebra and its Applications

#### Volume

436

#### Issue

9

#### Publisher

Elsevier

#### Publication Date

2011

#### First Page

3239

#### Last Page

3246

#### Abstract

The Boolean rank of an m×n(0,1)-matrix M is the minimum k for which matrices A and B exist with M=AB, A is m×k, B is k×n, and Boolean arithmetic is used. The intersection number of a directed graph D is the minimum cardinality of a finite set S for which each vertex v of D can be represented by an ordered pair (Sv,Tv) of subsets of S such that there is an arc from vertex u to vertex v in D if and only if Su∩Tv≠Ø. The intersection number of a digraph is equal to the Boolean rank of its adjacency matrix. Using this fact, we show that the intersection number of an upset tournament, equivalently, the Boolean rank of its adjacency matrix, is equal to the number of maximal subpaths of certain types in its upset path.

#### Recommended Citation

Brown, D. E., S. M. Roy, J. R. Lundgren, D. Siewert, Boolean rank of upset tournament matrices, Linear Algebra and its Applications, December (2011) DOI: 10.1016/j.laa.2011.11.003.