得分向量偏序集上Schur函数和奇异得分向量

Schur Function on the Poset of Score Vectors and Singular Score Vectors

所有 n维得分向量集合 Ln在优超关系下是一个偏序集。Ln上的实函数 g(s)称为 (严格 ) Schur凸的 ,若对任意 s,s′∈ L′n,s≠ s′,s优超 s′,恒有 g(s) (>) g(s′)。本文证明了 f (x) =s Ts和得分向量为 s的竞赛图Tn中 3-圈个数 c3 (s)在 Ln上分别是严格 Schur凸和严格 Schur凹的。称 n维得分向量 s为奇异的 ,若得分向量为 s的每个 n阶竞赛图 Tn的邻接矩阵都是奇异的。最后 ,应用 Ln上严格 Schur凸函数 f (s) 。

The set Ln of all n- term score vectors is a poset (i.e.,partially ordered set) under the majorization relation.A real function g(s) on the poset Lniscalled(strict) Schur convex if g(s) (>) g(s′) for any s,s′ ∈ Ln,s≠ s′,smajorizes s′.Itisproved thatf(s) =s Ts and c3 (s) ,the numberof 3- cycle in any tournament T with score vectorss,are strict Schur convex and strict Schur concave respectively.A n- term score vector s is called singular if every tournamentmatrix with score vectorssissingular.By using the strict Schurfunction f (s) on the poset Lnwe give a sufficient condition for a score vector being singular.