Geodesic
The Number of P-Vertices of Singular Acyclic Matrices: An Inverse Problem
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 525-532

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Let A be a real symmetric matrix. If after we delete a row and a column of the same index, the nullity increases by one, we call that index a P-vertex of A. When A is an n × n singular acyclic matrix, it is known that the maximum number of P-vertices is n − 2. If T is the underlying tree of A, we will show that for any integer number k ∈ 0, 1, . . ., n − 2, there is a (singular) matrix whose graph is T and with k P-vertices. We will provide illustrative examples.
Keywords: trees, acyclic matrices, singular, multiplicity of eigenvalues, P-set, P-vertices 
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Du, Zhibin; da Fonseca, Carlos M. The Number of P-Vertices of Singular Acyclic Matrices: An Inverse Problem. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 525-532. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a9/