Pick's inequality and tournaments

Abstract

Let T be a tournament on n nodes, and let A be its (adjacency) matrix. A. Brauer and I. Gentry observed that an inequality due to G. Pick implies that | Im λ|⩽ 1 2 cot( π 2n ) for all eigenvalues λ of A. We say that T is a Pick (or P -) tournament if equality holds for some λ. We determine when equality holds in Pick's inequality for arbitrary real matrices and use this to show that the P -tournaments can all be constructed from the transitive tournament M by reversing the arcs between the sets of certain node partitions or cuts { U, Ū}. The cuts are specified by ±1 n-vectors u such that u t w=0, where w=[ 1, σ,…, σ n− 1 ] t, σ=e iπ n . This links the cuts to cyclotomic polynomials. There is at least one P -tournament on n nodes if and only if n≠2 k , k⩾1. Up to isomorphism, there is precisely one P -tournament on n nodes if (and only if) n= p or n=2 p for some odd prime p. For odd n, up toisomorphism, the only regular P -tournament on n nodes has matrix Z n = Circ(0, 1,…, 1, 0,…, 0). A composition rule is used on the matrices Z p to form all P -tournaments on n=2 k p l nodes, l⩾1.