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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Nobuaki Obata
Graduate School of Information Sciences, Tohoku University, Sendai 980-8579 Japan.
Electrical & Electronics Engineering

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Distance matrices and quadratic embedding of graphs Nobuaki Obata; Alfi Y. Zakiyyah
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 6, No 1 (2018): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2018.6.1.4

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 nā€„ā‰¤ā€„5, among which two are not of QE class.
Co-Authors Alfi Y. Zakiyyah