# 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.

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