A Nordhaus–Gaddum conjecture for the minimum number of distinct eigenvalues of a graph

We propose a Nordhaus–Gaddum conjecture for q(G), the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph G: for every graph G excluding four exceptions, we conjecture that q(G)+q(Gc)≤|G|+2, where Gc is the complement of G. We compute q(Gc) for all trees and all gra...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 564; pp. 236 - 263
Main Authors Levene, Rupert H., Oblak, Polona, Šmigoc, Helena
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.03.2019
American Elsevier Company, Inc
Subjects
Eigenvalues
Graphs
Inverse eigenvalue problem for graphs
Linear algebra
Mathematical analysis
Matrix methods
Minimum number of distinct eigenvalues
Minimum rank
Nordhaus–Gaddum inequality
Orthogonal matrices
Trees
Orthogonal matrices
Minimum rank
05C50
15B10
Inverse eigenvalue problem for graphs
Minimum number of distinct eigenvalues
15B57
Nordhaus–Gaddum inequality
15A18
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Summary:We propose a Nordhaus–Gaddum conjecture for q(G), the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph G: for every graph G excluding four exceptions, we conjecture that q(G)+q(Gc)≤|G|+2, where Gc is the complement of G. We compute q(Gc) for all trees and all graphs G with q(G)=|G|−1, and hence we verify the conjecture for trees, unicyclic graphs, graphs with q(G)≤4, and for graphs with |G|≤7.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2018.12.001