Johnson, Charles R.Saiago, Carlos M.Toyonaga, Kenji2022-03-302022-12-172021-03-010024-3795PURE: 27629983PURE UUID: 433c4e7b-4311-498a-95ad-a6cc8b897bd5Scopus: 85097800652WOS: 000603481000007ORCID: /0000-0001-9843-3821/work/99432471http://hdl.handle.net/10362/135559In 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.18203089engCombinatorially symmetricEigenvalueGeometric multiplicityGraph of a matrixTreeAlgebra and Number TheoryNumerical AnalysisGeometry and TopologyDiscrete Mathematics and Combinatoricsjournal article10.1016/j.laa.2020.11.023https://www.scopus.com/pages/publications/85097800652