杜志斌. 具有极大P-集的实对称阵及其伴随图[J]. 华南师范大学学报(自然科学版), 2016, 48(1): 119-122.
引用本文: 杜志斌. 具有极大P-集的实对称阵及其伴随图[J]. 华南师范大学学报(自然科学版), 2016, 48(1): 119-122.
The real symmetric matrices with a P-set of maximum size and their associated graphs[J]. Journal of South China Normal University (Natural Science Edition), 2016, 48(1): 119-122.
Citation: The real symmetric matrices with a P-set of maximum size and their associated graphs[J]. Journal of South China Normal University (Natural Science Edition), 2016, 48(1): 119-122.

具有极大P-集的实对称阵及其伴随图

The real symmetric matrices with a P-set of maximum size and their associated graphs

  • 摘要: 实对称阵的P-集是一个基于矩阵的特征值重数以及Cauchy插值定理所提出的定义。设 为一个 阶实对称阵,记 为 的特征值0的(代数)重数,并记 为将 的第 行与第 列去掉后所得的主子阵,其中 为 的一个非空子集。特别地,当 时,称S为 的一个P-集。记 为实对称阵 的P-集所含元素个数的最大值。Kim与Shader证明了每个 阶实对称阵至多包含 个元素,即 。杜志斌与Fonseca首先将研究重点放在树矩阵(即伴随图为树的矩阵),研究了满足 的 阶树矩阵 ,并完全刻画出 的伴随图(树)。本文将研究范围从树矩阵延伸到所有实对称阵,研究了满足 的 阶实对称阵 ,给出其相关性质,并对 为偶数时 的伴随图进行特征刻画,而对 为奇数时 的伴随图给出了猜想,推广了关于树矩阵的结果。

     

    Abstract: The concept of P-set of real symmetric matrices is proposed based on the multiplicity of eigenvalues of matrices and Cauchy interlacing theory. Suppose that is a real symmetric matrix of order . Let be the multiplicity of eigenvalue 0 of , and let be the principal submatrix of obtained from by deleting the rows and columns, where is a nonempty subset of . In particular, when , we say S is a P-set of . Let be the maximum size among the P-sets of . Kim and Shader proved that every real symmetric matrix of order has at more elements, i.e., . Du and Fonseca first focused on the acyclic matrices (i.e., the matrices whose associated graphs are trees), investigated the acyclic matrices of order n satisfied , and completely characterized the associated graphs (trees) of .This paper extended the research from acyclic matrices to all real symmetric matrices, investigated the real symmetric matrices of order satisfied , presented their properties, and characterized the associated graphs when is even, conjectured the associated graphs when is odd, which improved the results on acyclic matrices.

     

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