Elsevier

Discrete Applied Mathematics

Volume 269, 30 September 2019, Pages 211-219
Discrete Applied Mathematics

The number of P-vertices for acyclic matrices of maximum nullity

Dedicated Prof. Domingos M. Cardoso in occasion of his 65th birthday
https://doi.org/10.1016/j.dam.2019.06.016Get rights and content
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Abstract

For a given n-by-n real symmetric matrix A, if the nullity of the principal submatrix of A, obtained from the deletion of a row and a column of the same index, goes up by one, we call such index P-vertex. In this note, we consider the problem of characterizing the trees for which there is an acyclic matrix of maximum nullity with an extremal number of P-vertices. The range of possible values for the number of P-vertices is determined as well. Some applications of aforementioned results and further properties on the so-called optimal P-sets are also discussed.

Keywords

Trees
Acyclic matrices
Multiplicity of eigenvalues
Maximum nullity
P-set
P-vertices

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