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.

Keywords

• eigenvalue
• graph of a matrix
• 2-downer edge cycle
• 15a18
• 05c50