The λ − τ structured inverse eigenvalue problem
Keivan Hassani Monfared, Bryan L. Shader · Linear and Multilinear Algebra · 2015
Let and be real numbers that satisfy the strict second-order Cauchy interlacing inequalities for and the nondegeneracy conditions for . Given a connected graph on vertices with adjacent vertices and , it is proven that there is a real symmetric matrix whose graph is such that has eigenvalues and has eigenvalues , provided some necessary combinatorial conditions on are satisfied. We also provide generalizations when the two deleted vertices are not adjacent, as well as interpretation of the results in terms of perturbing one or two diagonal entries.