Sets of Parter vertices which are Parter sets

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 gives 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 increases 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 paper 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.