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Distance spectra of graphs: a survey. (English) Zbl 1295.05093

Summary: R. L. Graham and H. O. Pollack [Bell Syst. Tech. J. 50, 2495–2519 (1971; Zbl 0228.94020)] 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.

MSC:

05C12 Distance in graphs
05C31 Graph polynomials
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C76 Graph operations (line graphs, products, etc.)
94C15 Applications of graph theory to circuits and networks

Citations:

Zbl 0228.94020
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Full Text: DOI

References:

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