SIAM J. Matrix Anal. Appl. 26 (2004/05), no. 2, 390–399.
Given an n-by-n Hermitian matrix A and a real number lambda, index i is said to be Parter (resp. neutral, downer) if the multiplicity of lambda as an eigenvalue of A(i) is one more (resp. the same, one less) than that in A. In case the multiplicity of lambda in A is at least 2 and the graph of A is a tree, there are always Parter vertices. Our purpose here is to advance the classification of vertices and, in particular, to relate classification to the combinatorial structure of eigenspaces. Some general results are given and then used to deduce some rather specific facts, not otherwise easily observed. Examples are given.