The maximum number of Parter vertices of acyclic matrices

Amélia Fonseca, Ângela Mestre, Ali Mohammadian, Cecília Perdigão, Maria Manuel Torres

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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.

Original languageEnglish
Article number112198
JournalDiscrete Mathematics
Issue number2
Publication statusPublished - Feb 2021


  • Acyclic matrix
  • Nullity
  • Parter vertex
  • Tree


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