Change in vertex status after removal of another vertex in the general setting

In the theory of multiplicities for eigenvalues symmetric matrices whose graph is a tree, it proved very useful to understand change in status (Parter, neutral, or downer) one vertex upon removal another given (both case two vertices are adjacent non-adjacent). As subject has evolved toward study more general matrices, over fields, with graphs, appropriate resolve same type question settings. “Multiplicity” now means geometric multiplicity. Here, we give complete resolution three settings and compare these classical (216 “Yes” “No” results). consequence, several unexpected insights recorded.