On unimodular tournaments

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 \(n - \lfloor\log_2(n)\rfloor\) vertices.