Distance matrices and quadratic embedding of graphs
Abstract
A connected graph is said to be of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently, if the distance matrix is conditionally negative definite. Several criteria for a graph to be of QE class are derived from the point of view of graph operations. For a quantitative criterion the QE constant is introduced and concrete examples are shown with explicit calculation. If the distance matrix admits a constant row sum, the QE constant coincides with the second largest eigenvalue of the distance matrix. The QE constants are determined for all graphs on n vertices with n ≤ 5, among which two are not of QE class.
Keywords
conditionally negative definite matrix, distance matrix, Euclidean distance matrix quadratic embedding, QE constant
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PDFDOI: http://dx.doi.org/10.5614/ejgta.2018.6.1.4
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ISSN: 2338-2287
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