# Matricial characterization of tournaments with maximum number of diamonds

## TL;DR

This paper provides a complete matrix-based characterization of n-tournaments with the maximum number of diamonds for certain sizes, assuming skew-conference matrices, and offers bounds for other sizes.

## Contribution

It introduces a matricial framework to characterize maximum diamond tournaments for specific sizes, extending understanding of their structure.

## Key findings

- Complete characterization for n ≡ 0 mod 4 and n ≡ 3 mod 4
- Upper bounds and matricial descriptions for n ≡ 2 mod 4
- Assumption of skew-conference matrices

## Abstract

A diamond is a $4$-tournament which consists of a vertex dominating or dominated by a $3$-cycle. Assuming the existence of skew-conference matrices, we give a complete characterization of $n$-tournaments with the maximum number of diamonds when $n\equiv0\pmod{4}$ and $n\equiv3\pmod{4}$. For $n\equiv2\pmod{4}$, we obtain an upper bound on the number of diamonds in an $n$-tournament and we give a matricial characterization of tournaments achieving this bound.