  It is shown how a connected graph and a tree with partially prescribed
spectrum can be constructed. These constructions are based on a recent result
of Salez that every totally real algebraic integer is an eigenvalue of a tree.
Our result implies that for any (not necessarily connected) graph $G$, there is
a tree $T$ such that the characteristic polynomial $P(G,x)$ of $G$ can divide
the characteristic polynomial $P(T,x)$ of $T$, i.e., $P(G,x)$ is a divisor of
$P(T,x)$.
