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The maximum number of Parter vertices of acyclic matrices
Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.disc.2020.112198
Amélia Fonseca , Ângela Mestre , Ali Mohammadian , Cecília Perdigão , Maria Manuel Torres

Abstract A vertex v of the underlying graph of a symmetric matrix A is called ‘Parter’ if the nullity of the matrix obtained from A by removing the row and column indexed by v is more than the nullity of A . Let A be a singular symmetric matrix with rank r whose underlying graph is a tree. It is known that the number of Parter vertices of A is at most r − 1 . We prove that when r is odd this number is at most r − 2 . We characterize the trees where these bounds are achieved.

中文翻译:

无环矩阵Parter顶点的最大数目

摘要 如果通过去除由 v 索引的行和列从 A 获得的矩阵的无效性大于 A 的无效性,则对称矩阵 A 的底层图的顶点 v 称为“Parter”。设 A 是一个奇异对称矩阵,秩为 r,其底层图是一棵树。已知A 的Parter 顶点数最多为r − 1 。我们证明当 r 为奇数时,这个数至多为 r − 2 。我们描述了实现这些边界的树。
更新日期:2021-02-01
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