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: $+1$ (Parter); no change (neutral); and $-1$ (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.

域上方矩阵图中的顶点可以根据它们的移除如何改变已识别特征值的几何多重性来分类。存在三种可能性：$+\mathrm{1个}$（合伙人）；没有变化（中性）；和$-\mathrm{1个}$（沮丧）。当图是一棵树时，“下分支机制”区分 Parter 顶点。在这里，我们发现了这种机制如何推广到一般图，包括 Hermitian 矩阵和一般矩阵。然后，我们应用新思想，既对一般图中的悬垂边进行分类，又在存在 2-downer 边时理解一般图中 2-downer 边循环的存在。这是对为什么这样的边不能出现在树中的进一步解释。