Distance spectra of graphs: A survey

In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Sinc...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 458; pp. 301 - 386
Main Authors Aouchiche, Mustapha, Hansen, Pierre
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.10.2014
Subjects
Characteristic polynomial
Distance matrix
Eigenvalues
Graph
Largest eigenvalue
05C76
05C31
05C12
Graph
05C50
Characteristic polynomial
Eigenvalues
Largest eigenvalue
Distance matrix
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Summary:In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2014.06.010