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

As a variant of the Ulam’s vertex reconstruction conjecture and the Harary’s edge reconstruction conjecture, Cvetković 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 $\{G-v\mathrm{\mid }v\in V\}$ for $\mathrm{\mid }V\mathrm{\mid }\ge 3$? 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 $\{G-e\mathrm{\mid }e\in E\}$? In this paper, we prove that if $\mathrm{\mid }V\mathrm{\mid }\mathrm{\ne }\mathrm{\mid }E\mathrm{\mid }$, then the characteristic polynomial of $G$ can be reconstructed from the characteristic polynomials of all subgraphs in $\{G-uv,G-u-v\mathrm{\mid }uv\in E\}$, 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 $\{G-e\mathrm{\mid }e\in E\}$ (resp. if $\mathrm{\mid }V\mathrm{\mid }\mathrm{\ne }\mathrm{\mid }E\mathrm{\mid }$).