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Pick's inequality and tournaments

By D.A. Gregory, S.J. Kirkland and B.L. Shader

Abstract

AbstractLet 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|⩽12cot(π2n) 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 utw=0, where w=[1, σ,…, σn−1]t, σ=eiπn. This links the cuts to cyclotomic polynomials. There is at least one P-tournament on n nodes if and only if n≠2k, k⩾1. Up to isomorphism, there is precisely one P-tournament on n nodes if (and only if) n=p or n=2p for some odd prime p. For odd n, up toisomorphism, the only regular P-tournament on n nodes has matrix Zn = Circ(0, 1,…, 1, 0,…, 0). A composition rule is used on the matrices Zp to form all P-tournaments on n=2kpl nodes, l⩾1

Publisher: Published by Elsevier Inc.
Year: 1993
DOI identifier: 10.1016/0024-3795(93)90281-R
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