Local Out-Tournaments with Upset Tournament Strong Components \(I\): Full and Equal \(\{0,1\}\)-Matrix Ranks

Abstract

A digraph \(D\) is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, \(n\). Corresponding results for biclique covers and partitions of the digraph are provided.