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arXiv · 2109.11809

On unimodular tournaments

Wiam Belkouche·Abderrahim Boussaïri·Abdelhak Chaïchaâ·Soufiane Lakhlifi
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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.

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Wiam Belkouche, Abderrahim Boussaïri, Abdelhak Chaïchaâ, Soufiane Lakhlifi. 2021-09-24. On unimodular tournaments. https://doi.org/10.1016/j.laa.2021.09.014

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