On unimodular tournaments

Abstract

A tournament is unimodular if the determinant of its skew-adjacency matrix is 1. In this paper, we give some properties and constructions of unimodular tournaments. A unimodular tournament T with skew-adjacency matrix S is invertible if S −1 is the skew-adjacency matrix of a tournament. A spectral characterization of invertible tournaments is given. Lastly, we show that every n-tournament can be embedded in a unimodular tournament by adding at most n − ⌊log 2 (n)⌋ vertices.