Distance matrices and quadratic embedding of graphs

Nobuaki Obata, Alfi Y. Zakiyyah

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)37-60
Number of pages24
JournalElectronic Journal of Graph Theory and Applications
Issue number1
Publication statusPublished - 2018


  • Conditionally negative definite matrix
  • Distance matrix
  • Euclidean distance matrix quadratic embedding
  • QE constant


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