An inverse eigenvalue problem for structured matrices determined by graph pairs (Q6615438)

Let \( A, B \in \mathbb{R}^{n \times n} \) be a pair of real symmetric matrices, where the nonzero patterns are determined by the edges of a given pair of graphs on \( n \) vertices.\N\NThe primary result of this paper concerns the following conjecture:\N\N{Conjecture.} If \( G \) and \( H \) are any chosen vertex-labeled graphs, then \( S(G, H) \) permits an arbitrary spectrum for the structured matrix \( C \) defined as\N\[\NC = \begin{pmatrix} A & B \\\NI & O \end{pmatrix} \in \mathbb{R}^{2n \times 2n}\,,\N\]\Nwhere, for vertex-labeled graphs \( G \) and \( H \) on \( n \geq 1 \) vertices,\N\[\NS(G, H) = \{ C(A, B) \in \mathbb{R}^{2n \times 2n} \mid A \in S(G), B \in S(H) \}.\N\]\N\NThe paper presents proofs of this conjecture under different assumptions on the graphs \( G \) and \( H \), which rely on a structured Jacobian method for matrices \( A \) and \( B \) of order at most 4, or when the graph associated with \( A \) contains a Hamiltonian path.\N\NFurthermore, a weaker version of this conjecture is established for any pair of graphs, with a restriction placed on the multiplicities of the eigenvalues of \( C \).