TY - JOUR
T1 - Fundamental graphs for the maximum multiplicity of an eigenvalue among Hermitian matrices with a given graph
AU - Johnson, Charles R.
AU - Leal-Duarte, António
AU - Saiago, Carlos M.
N1 - info:eu-repo/grantAgreement/FCT/Concurso de avaliação no âmbito do Programa Plurianual de Financiamento de Unidades de I&D (2017%2F2018) - Financiamento Base/UIDB%2F00297%2F2020/PT#
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDP%2F00297%2F2020/PT#
Funding Information:
Open access funding provided by FCT|FCCN (b-on). The work of the corresponding author is funded by national funds through the FCT-Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications).
Publisher Copyright:
© The Author(s) 2025.
PY - 2025/1/27
Y1 - 2025/1/27
N2 - Our purpose is to identify the graphs that are “fundamental” for the maximum multiplicity problem for Hermitian matrices with a given undirected simple graph. Like paths for trees, these are the special graphs to which the maximum multiplicity problem may be reduced. These are the graphs for which maximum multiplicity implies that all vertices are downers. Examples include cycles and complete graphs, and several more are identified, using the theory developed herein. All the unicyclic graphs that are fundamental, are explicitly identified. We also list those graphs with two edges added to a tree, and their maximum multiplicities, which we have found so far to be fundamental. A formula for maximum multiplicity is given based on fundamental graphs.
AB - Our purpose is to identify the graphs that are “fundamental” for the maximum multiplicity problem for Hermitian matrices with a given undirected simple graph. Like paths for trees, these are the special graphs to which the maximum multiplicity problem may be reduced. These are the graphs for which maximum multiplicity implies that all vertices are downers. Examples include cycles and complete graphs, and several more are identified, using the theory developed herein. All the unicyclic graphs that are fundamental, are explicitly identified. We also list those graphs with two edges added to a tree, and their maximum multiplicities, which we have found so far to be fundamental. A formula for maximum multiplicity is given based on fundamental graphs.
KW - Eigenvalue multiplicity
KW - Fundamental graph
KW - Graph of an Hermitian matrix
KW - Maximum multiplicity
KW - Trees
UR - http://www.scopus.com/inward/record.url?scp=85218119572&partnerID=8YFLogxK
UR - https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=nova_api&SrcAuth=WosAPI&KeyUT=WOS:001408072100001&DestLinkType=FullRecord&DestApp=WOS_CPL
UR - https://www.webofscience.com/wos/woscc/full-record/WOS:001408072100001
U2 - 10.1007/s43036-025-00420-6
DO - 10.1007/s43036-025-00420-6
M3 - Article
AN - SCOPUS:85218119572
SN - 2538-225X
VL - 10
SP - 1
EP - 8
JO - Advances in Operator Theory
JF - Advances in Operator Theory
M1 - 30
ER -