Let $P_G$ denote the matching generating function of a finite simple bipartite graph $G$.
Let now $H$ be a $2$-lift of $G$. We know (see for example Proposition 5.3.3 in Barvinok's book Combinatorics and Complexity of Partition Functions) that
$$ P_H(t)\le P_G(t)^2\quad \forall t\ge 0. $$
I was wondering if anything interesting could be said about the relation between the roots of $P_H(t)$ and the roots of $P_G(t)$.
More generally, what can be said about $P_H(t)$ in terms of $P_G(t)$ other than the above statement?
