  We consider matrices of the form $qD+A$, with $D$ being the diagonal matrix
of degrees, $A$ being the adjacency matrix, and $q$ a fixed value. Given a
graph $H$ and $B\subseteq V(G)$, which we call a coalescent pair $(H,B)$, we
derive a formula for the characteristic polynomial where a copy of same rooted
graph $G$ is attached by the root to \emph{each} vertex of $B$. Moreover, we
establish if $(H_1,B_1)$ and $(H_2,B_2)$ are two coalescent pairs which are
cospectral for any possible rooted graph $G$, then $(H_1,V(H_1)\setminus B_1)$
and $(H_2,V(H_2)\setminus B_2)$ will also always be cospectral for any possible
rooted graph $G$.
