Distance matrices and quadratic embedding of graphs

Nobuaki Obata (Graduate School of Information Sciences, Tohoku University, Sendai 980-8579 Japan.)
Alfi Y. Zakiyyah (Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Ganesa 10 Bandung 40132, Indonesia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 6, No 1 (2018): Electronic Journal of Graph Theory and Applications

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.