### 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.

Original language | English |
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Pages (from-to) | 37-60 |

Number of pages | 24 |

Journal | Electronic Journal of Graph Theory and Applications |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

Obata, N., & Zakiyyah, A. Y. (2018). Distance matrices and quadratic embedding of graphs.

*Electronic Journal of Graph Theory and Applications*,*6*(1), 37-60. https://doi.org/10.5614/ejgta.2018.6.1.4