Pick's inequality and tournaments (Q1801444)

If \(\lambda\) is an eigenvalue of the adjacency matrix of a tournament \(T_ n\) with \(n\) nodes, then \(|\text{Im} \lambda|\leq(1/2)\text{cot}(\pi/2n)\). The authors show, among other things, that all tournaments for which equality hold for some \(\lambda\) can be constructed from transitive tournaments by certain cut-reversals and that, in particular, such tournaments \(T_ n\) exist if and only if \(n\neq 2^ k\), \(k\geq 1\).