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

• Autor(a) principal: Johnson, Charles R.
• Data de Publicação: 2021
• Outros Autores: Saiago, Carlos M., Toyonaga, Kenji
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10362/135559
• topic: Combinatorially symmetric; Eigenvalue; Geometric multiplicity; Graph of a matrix; Tree; Algebra and Number Theory; Numerical Analysis; Geometry and Topology; Discrete Mathematics and Combinatorics

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