The effect of removing a 2-downer edge or a cut 2-downer edge triangle for an eigenvalue

Abstract Edges in the graph associated with a square matrix over field may be classified as to how their removal affects multiplicity of an identified eigenvalue. There are five possibilities: + 2 +2 (2-Parter); 1 +1 (Parter); no change (neutral); − -1 (downer); and -2 (2-downer). Especially, it is known that 2-downer edges for eigenvalue comprise cycles graph. We investigate effect statuses other or vertices by removing edge. Then, we cut edge triangle.