Recovering the characteristic polynomial of a graph from entries of the adjugate matrix

The adjugate matrix of $G$, denoted by $\operatorname{adj}(G)$, is the adjugate of the matrix $x\mathbf{I}-\mathbf{A}$, where $\mathbf{A}$ is the adjacency matrix of $G$. The polynomial reconstruction problem (PRP) asks if the characteristic polynomial of a graph $G$ can always be recovered from the multiset $\operatorname{\mathcal{PD}}(G)$ containing the $n$ characteristic polynomials of the vertex-deleted subgraphs of $G$. Noting that the $n$ diagonal entries of $\operatorname{adj}(G)$ are precisely the elements of $\operatorname{\mathcal{PD}}(G)$, we investigate variants of the PRP in which multisets containing entries from $\operatorname{adj}(G)$ successfully reconstruct the characteristic polynomial of $G$. Furthermore, we interpret the entries off the diagonal of $\operatorname{adj}(G)$ in terms of characteristic polynomials of graphs, allowing us to solve versions of the PRP that utilize alternative multisets to $\operatorname{\mathcal{PD}}(G)$ containing polynomials related to characteristic polynomials of graphs, rather than entries from $\operatorname{adj}(G)$.