DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory

Article in press

Authors:

Z. Du, C.M. da Fonseca

Title:

The number of P-vertices of singular acyclic matrices: An inverse problem

Source:

Discussiones Mathematicae Graph Theory

Received: 2019-02-14, Revised: 2019-09-05, Accepted: 2019-11-06, https://doi.org/10.7151/dmgt.2282

Abstract:

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 $n\times n$ 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 $k\in\{0,1,\ldots,n-2\}$, 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