Joining Forces for Reconstruction Inverse Problems

A spectral inverse problem concerns the reconstruction of parameters a parent graph from prescribed data subgraphs. Also referred to as P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, aim is seek sets determine uniquely. Other related problems, including Polynomial Problem (PRP), involve recovery invariants. The PRP seeks extract spectrum deck cards each showing vertex-deleted subgraph. We show how various algebraic methods join forces reconstruct its invariants minimal set restricted eigenvalue-eigenvector information functions entries eigenvectors adjacency matrix can be retrieved eigenvalues A. establish that there are two subclasses disconnected graphs with card common eigenvalue. These could occur possible counter examples positive solution PRP.