Distance matrices and quadratic embedding of graphs

Nobuaki Obata, Alfi Y. Zakiyyah

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 $n\le5$, 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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DOI: http://dx.doi.org/10.5614/ejgta.2018.6.1.4

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