Parter vertices and generalization of the downer branch mechanism in the general setting

Abstract

Vertices in the graph of a square matrix over a field may be classified as to how their removal changes the geometric multiplicity of an identified eigenvalue. There are three possibilities: (Parter); no change (neutral); and (downer). When the graph is a tree, the ‘downer branch mechanism’ distinguishes the Parter vertices. Here, we discover how this mechanism generalizes for general graphs, both for Hermitian matrices and general matrices. Then, we apply the new ideas, both to classify pendent edges in general graphs, and to understand the existence of 2-downer edge cycles in general graphs, when there is a 2-downer edge. This is a further explanation of why such edges cannot occur in trees.