On unimodular tournaments

TL;DR

This paper investigates unimodular tournaments, exploring their properties, constructions, spectral characteristics, and how any tournament can be embedded into a unimodular one with minimal added vertices.

Contribution

It provides new properties, a spectral characterization of invertible unimodular tournaments, and a method to embed any tournament into a unimodular tournament with few additional vertices.

Findings

Unimodular tournaments have specific algebraic properties.

Spectral characterization links invertibility to skew-adjacency matrices.

Any tournament can be embedded into a unimodular tournament with at most n - floor(log2(n)) vertices.

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