具有极谱的二部竞赛矩阵

The Bipartite Tournament Matrix with Extreme Spectra

令Γm ,n 表示所有的不可约m×n二部竞赛矩阵 ,获得了如下主要结论 :(1)Γm ,n 中每个 (s,t)半正则二部竞赛矩阵的特征值的代数重数和几何重数相等 ;(2 )刻划了Γm ,n 中恰好有四个不同特征值的 (s ,t) 半正则二部竞赛矩阵 ,这些矩阵与组合设计有关 ;(3)设 lm ,n 表示 Γm ,n 中零特征值的最大代数重数 ,则lm ,n =m +n - 4 ,并给出了使该式成立的二部竞赛图的结构。

Let Γ m,n  be the set of irreducible m×n bipartite tournament matrix. Here are main results. (1) Every eigenvalue of belonging (s,t) semi-regular bipartite tournament matrix in Γ m,n  has the same algebraic and geometric multiplicity. (2) we characterize the (s,t) semi-regular bipartite tournament matrices in Γ m,n  having exactly four distinct eigenvalues, which are relating to combinatory design. (3) Let β m,n  be the maximum algebraic multiplicity of 0 as an eigenvalue of the matrix in Γ m,n , then β m,n =m+n-4 and we give all m×n bipartite tournaments that algebraic multiplicity of 0 as an eigenvalue of adjacency matrices in m+n-4