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

<jats:title>Abstract</jats:title> <jats:p>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: <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2022-0186_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:math> <jats:tex-math>+2</jats:tex-math> </jats:alternatives> </jats:inline-formula> (2-Parter); <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2022-0186_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>+1</jats:tex-math> </jats:alternatives> </jats:inline-formula> (Parter); no change (neutral); <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2022-0186_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>-1</jats:tex-math> </jats:alternatives> </jats:inline-formula> (downer); and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2022-0186_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:math> <jats:tex-math>-2</jats:tex-math> </jats:alternatives> </jats:inline-formula> (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.</jats:p>