Article

Boolean Rank of Upset Tournament Matrices

Linear Algebra and its Applications
Document Type

Article
Publisher

Elsevier
Publication Date

1-1-2011
DOI

10.1016/j.laa.2011.11.003
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.

Citation Information

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.