Determinants of Seidel tournament matrices

Abstract

The Seidel matrix of a tournament on n players is an $n\times n$ 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 $n\times n$ 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, $D\left(n\right)$, of square roots of determinants of $n\times n$ Seidel matrices. It is shown that $D\left(n\right)$ 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 $D\left(n\right)$ for n even. The expected value and variance of $\det S$ over the $n\times n$ Seidel matrices chosen uniformly at random is determined, and upper bounds on $\max D\left(n\right)$ are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many n, $D\left(n\right)$ contains a gap (that is, there are odd integers $k<\ell <m$ such that $k,m\in D\left(n\right)$ but $\ell \notin D\left(n\right)$) and several properties of the characteristic polynomials of Seidel matrices are established.