Let $Q$ be a finite connected and directed graph with $n$ points. Assume $Q$ is acyclic as a directed graph. Let $C=C_Q$ be the Cartan matrix of $Q$, that is the matrix with entries $c_{i,j}$ being the number of paths from $j$ to $i$. Let $\phi_Q$ be the Coxeter matrix of $Q$ defined as $\phi_Q=-C^{-1} C^T$ and $g_Q$ the Coxeter polynomial defined as the characteristic polynomial of $\phi_Q$.
Two such graphs $Q_1$ and $Q_2$ are said to be derived equivalent when the path algebras $KQ_1$ and $KQ_2$ are derived equivalent or equivalently when $Q_2$ is obtainable from $Q_2$ by switching orientations of arrows (so that the graphs stay acyclic). Being derived equivalent implies that the Coxeter polynomials are the same.
Question 1: Assume we have a polynomial $p$ and know that it is the Coxeter polynomial of some graph $Q$. Is there a way (or a software) to obtain such a graph $Q$ from the given polynomial in an easy way?
Question 2: For which Coxeter polynomials $p_Q$ is it true that that $p_Q=p_{Q'}$ for some other graph $Q'$ if and only if $Q$ and $Q'$ are derived equivalent?
(this is for example true for Coxeter polynomials coming from (affine) Dynkin types)
