Determinants of Seidel Tournament Matrices

TL;DR

This paper investigates the set of determinants of Seidel matrices associated with tournaments, exploring their properties, bounds, and distribution, and relating findings to the Hadamard conjecture.

Contribution

It characterizes the set of possible determinants, establishes bounds, and uncovers gaps in the set for infinitely many sizes, advancing understanding of Seidel matrices in combinatorics.

Findings

$ ext{det } S=0$ for odd $n$

Every odd integer in $[1, 1+n^2/2]$ is in $ ext{Det}(n)$ for even $n$

Identifies gaps in the set $ ext{Det}(n)$ for infinitely many $n$

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 \[ \mathcal{D}(n)= \{ \sqrt{\det S}: \mbox{ $S$ is an $n\times n$ Seidel matrix}\}. \] This paper studies various questions about $\mathcal{D}(n)$. It is shown that $\mathcal{D}(n)$ is a proper subset of $\mathcal{D}(n+2)$ for every positive even integer, and every odd integer in the interval $[1, 1+n^2/2]$ is in $\mathcal{D}(n)$ 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 $\max \mathcal{D}(n)$ are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many $n$, $\mathcal{D}(n)$ contains a gap (that is, there are odd integers $k<\ell <m$ such that $k, m \in \mathcal{D}(n)$ but $\ell \notin \mathcal{D}(n)$) and several properties of the characteristic polynomials of Seidel matrices are established.