An inverse eigenvalue problem for structured matrices determined by graph pairs

Given a pair of real symmetric matrices $A,B\in R^{n\times n}$ with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix $C=\left[\begin{array}{cc}A& B\\ I& O\end{array}\right]\in R^{2n\times 2n}$. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C.