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On unimodular tournaments. (English) Zbl 1478.05064

Summary: 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.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)

Software:

SageMath
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References:

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