On the edge reconstruction of the characteristic and permanental polynomials of a simple graph

As a variant of the Ulam-Kelly's vertex reconstruction conjecture and the Harary's edge reconstruction conjecture, Cvetkovie and Schwenk posed independently the following problem: can the characteristic polynomial of a simple graph G with vertex set V be reconstructed from the characteristic polynomials of all subgraphs in {G - v | v is an element of V} for |V| >= 3? This problem is still open. A natural problem is: can the characteristic polynomial of a simple graph G with edge set E be reconstructed from the characteristic polynomials of all subgraphs in {G - e|e is an element of E}? In this paper, we prove that if |V| not equal |E|, then the characteristic polynomial of G can be reconstructed from the characteristic polynomials of all subgraphs in {G - uv, G - u - v|uv is an element of E}, and the similar result holds for the permanental polynomial of G. We also prove that the Laplacian (resp. signless Laplacian) characteristic polynomial of G can be reconstructed from the Laplacian (resp. signless Laplacian) characteristic polynomials of all subgraphs in {G - e|e is an element of E} (resp. if |V| not equal |E|). (c) 2024 Elsevier B.V. All rights reserved.