  The quadratic embedding constant (QEC) of a finite, simple, connected graph
$G$ is the maximum of the quadratic form of the distance matrix of $G$ on the
subset of the unit sphere orthogonal to the all-ones vector. The study of these
QECs was motivated by the classical work of Schoenberg on quadratic embedding
of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In
this article, we provide sharp upper and lower bounds for the QEC of trees. We
next explore the relation between distance spectra and quadratic embedding
constants of graphs - and show two further results: $(i)$ We show that the
quadratic embedding constant of a graph is zero if and only if its second
largest distance eigenvalue is zero. $(ii)$ We identify a new subclass of
nonsingular graphs whose QEC is the second largest distance eigenvalue.
Finally, we show that the QEC of the cluster of an arbitrary graph $G$ with
either a complete or star graph can be computed in terms of the QEC of $G$. As
an application of this result, we provide new families of examples of graphs of
QE class.
