| Summary: | <p>A connected graph is said to be of <span class="math"><em>Q</em><em>E</em></span> 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 <span class="math"><em>Q</em><em>E</em></span> class are derived from the point of view of graph operations. For a quantitative criterion the <span class="math"><em>Q</em><em>E</em></span> constant is introduced and concrete examples are shown with explicit calculation. If the distance matrix admits a constant row sum, the <span class="math"><em>Q</em><em>E</em></span> constant coincides with the second largest eigenvalue of the distance matrix. The <span class="math"><em>Q</em><em>E</em></span> constants are determined for all graphs on <span class="math"><em>n</em></span> vertices with <span class="math"><em>n</em> ≤ 5</span>, among which two are not of <span class="math"><em>Q</em><em>E</em></span> class.</p>
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