Structure of singular and nonsingular tournament matrices

TL;DR

This paper investigates the relationship between the cycle structure of tournaments and the singularity of their adjacency matrices, providing precise counts of three-cycle configurations that determine matrix singularity.

Contribution

It offers a detailed classification of tournaments based on the number of three-cycles and links this to the singularity of their adjacency matrices.

Findings

Number of three-cycles determines matrix singularity.

Structural classifications of tournaments with specific three-cycle counts.

Explicit criteria for singular and nonsingular tournament matrices.

Abstract

A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a matrix with $0$s on its diagonal and all other entries equal to $1$. An outstanding question in tournament theory asks to classify the adjacency matrices of tournaments which are singular (or nonsingular). We study this question using the structure of tournaments as graphs, in particular their cycle structure. More specifically, we find, as precisely as possible, the number of cycles of length three that dictates whether the corresponding tournament matrix is singular or nonsingular. We also give structural classifications of the tournaments that have the specified numbers of cycles of length three.