Determinants of Seidel tournament matrices

The Seidel matrix of a tournament on n players is an n x n skew-symmetric matrix with entries in {0, 1, -1} that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an n x n Seidel matrix is 0 if n is odd, and is an odd perfect square if n is even. This leads to the study of the set, D ( n ), of square roots of determinants of n x n Seidel matrices. It is shown that D ( n ) is a proper subset of D(n + 2) for every positive even integer, and every odd integer in the interval [1, 1 + n (2) / 2 ] is in D ( n ) for n even. The expected value and variance of det S over the n x n Seidel matrices chosen uniformly at random is determined, and upper bounds on max D ( n ) are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many n , D ( n ) contains a gap (that is, there are odd integers k < l < m such that k, m is an element of D ( n ) but l is not an element of/ D ( n )) and several properties of the characteristic polynomials of Seidel matrices are established. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.