Quadrangularity and Strong Quadrangularity in Tournaments
  The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero entries of M with a 1. A directed graph is said to support M if its adjacency matrix is the pattern of M. If M is an orthogonal matrix, then a digraph which supports M must satisfy a condition known as quadrangularity. We look at quadrangularity in tournaments and determine for which orders quadrangular tournaments exist. We also look at a more restrictive necessary condition for a digraph to support an orthogonal matrix, and give a construction for tournaments which meet this condition. 
J. Richard Lundgren (CU-Denver), K.B. Reid (CSUSM), Simone Severini   (Univ. of York), Dustin J. Stewart (CU-Denver)