Changes in vertex status and the fundamental decomposition of a tree relative to a multiple (parter) eigenvalue

Given a tree T, a Hermitian matrix A whose graph is T and an eigenvalue λ of A, a number of new structural results about T relative to the multiplicity of λ in A are developed. These include a complete classification of the possible changes in status of one vertex upon removal of another. These are used, in part, to give a unique fundamental decomposition of the tree that can be used to answer further structural questions. In the process, the notions of singly and multiply Parter vertices are introduced and used. Possible changes in the structure of the fundamental decomposition, resulting from changes in one diagonal entry, or deletion of a row and a column, are also discussed.