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