Eigenvalue-free intervals of distance matrices of threshold and chain graphs

We prove that for two graph classes, threshold graphs and chain graphs, there exist intervals in which their distance matrix does not have any eigenvalues. We also compute the determinant of the distance matrix of both threshold and chain graphs in terms of their generating binary sequence. In this...

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Bibliographic Details
Published in:Linear & multilinear algebra 2021-12, Vol.69 (16), p.2959-2975
Main Authors: Alazemi, Abdullah, Anđelić, Milica, Koledin, Tamara, Stanić, Zoran
Format: Article
Language:English
Subjects:
chain graph
Chains
Distance matrix
eigenvalue-free interval
Eigenvalues
Graphs
Intervals
treshold graph
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Summary:We prove that for two graph classes, threshold graphs and chain graphs, there exist intervals in which their distance matrix does not have any eigenvalues. We also compute the determinant of the distance matrix of both threshold and chain graphs in terms of their generating binary sequence. In this way we positively address a research problem posed in [D.P. Jacobs, V. Trevisan, F. C. Tura, Distance eigenvalue location in thresholds graphs, Proceedings of DGA, Manaus; 2013. p. 1-4].
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2019.1701624