TY - JOUR

T1 - Change in vertex status after removal of another vertex in the general setting

AU - Johnson, Charles R.

AU - Saiago, Carlos M.

AU - Toyonaga, Kenji

N1 - info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT#

PY - 2021/3/1

Y1 - 2021/3/1

N2 - 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.

AB - 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.

KW - Combinatorially symmetric

KW - Eigenvalue

KW - Geometric multiplicity

KW - Graph of a matrix

KW - Tree

UR - http://www.scopus.com/inward/record.url?scp=85097800652&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.11.023

DO - 10.1016/j.laa.2020.11.023

M3 - Article

AN - SCOPUS:85097800652

VL - 612

SP - 128

EP - 145

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -