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.