Henrique Cruz
Universidade da Beira Interior

Abstract: Given an Hermitian matrix, whose graph is a tree, having a multiple eigenvalue $\lambda$, the Parter-Wiener theorem guarantees the existence of principal submatrices for which the multiplicity of $\lambda$ increases. The vertices of the tree whose removal give rise to these principal submatrices are called weak Parter vertices and with some additional conditions are called Parter vertices. A set of $k$ Parter vertices whose removal increase the multiplicity of $\lambda$ by $k$ is called Parter set. As observed by several authors a set of Parter vertices is not necessarily a Parter set. In this talk we present some results on Parter vertices and on Parter sets. We prove that if $A$ is a symmetric matrix, whose graph is a tree, and $\lambda$ is an eigenvalue of $A$ whose multiplicity does not exceed $3$, then every set of Parter vertices, for $\lambda$ relative to $A$, is also a Parter set. We also present an upper and a lower bound of the number of weak Parter vertices, for an eigenvalue $\lambda$ of a matrix $A$ whose graph is a tree, with the assumption that $\lambda$ has maximum multiplicity.
This is a joint work with Rosário Fernandes.

Seminário financiado por Fundos Nacionais através da FCT – Fundação para a Ciência e Tecnologia no âmbito do projeto UID/MAT/04721/2013.