Boolean Rank and Isolation Number of N-Regular Tournaments

We examine Boolean rank and isolation number of a class of matrices, the adjacency matrices of regular tournaments. Boolean rank is defined as the minimum k such that a m x n matrix can be factored into m x k and k x n matrices, using Boolean arithmetic. Isolation number is defined as the maximum number of 1’s that do not share a row, column, or 2 x 2 submatrix of 1’s. Linear programming can be applied by using the underlying structure of the tournament matrices to develop a relationship between Boolean rank and isolation number. We show possible methods for relating the two more strongly using the biclique matrix. A biclique matrix, B, is a matrix such that the rows are indexed by directed bicliques and columns are indexed by arcs such that bij = 1 if and only if arc i is contained in directed biclique j. Furthermore, we present documented code that analyzes individual cases computationally. Finally, we display specific cases generated by the code that display a Boolean rank less than the vertex count