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Fundamental graphs for the maximum multiplicity of an eigenvalue among Hermitian matrices with a given graph
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Resumo(s)
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.
Descrição
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.
Palavras-chave
Eigenvalue multiplicity Fundamental graph Graph of an Hermitian matrix Maximum multiplicity Trees Analysis Algebra and Number Theory