Matricial characterization of tournaments with maximum number of diamonds

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 $n\equiv 0(\mathrm{mod}4)$ and $n\equiv 3(\mathrm{mod}4)$. For $n\equiv 2(\mathrm{mod}4)$, 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.