Abstract
A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.
Highlights
An undirected graph G has a vertex set V ( G ) = {1, 2, . . . , n} of n vertices and an edge set E joining pairs of the vertices
If the graphs in a pair ( H, G ), of graphs with the same p-deck have different characteristic polynomial, the pair would be a counter example to the positive resolution of the Polynomial Reconstruction Problem (PRP)
We have proved that potential counter examples to the PRP may occur in both Classes
Summary
A p-deck provides sufficient information for the parent graph to be constructed, reconstruction follows If both stages are performed successfully, the graphs in class C are said to be polynomial reconstructible and the PRP is said to be solved positively for C. Another approach, mainly used regarding the PRP, is the counter example technique [7,12]. Deleted subgraph from the eigenspaces of the parent graph leading to new proofs of two well known theorems, namely Clarke’s derivative of the characteristic polynomial and Cauchy’s Inequalities for non-negative matrices We give a characterization of the graphs in the two classes
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