On the edge reconstruction of the characteristic and permanental polynomials of a simple graph (2310.07104v1)

Abstract: As a variant of the Ulam's vertex reconstruction conjecture and the Harary's edge reconstruction conjecture, Cvetkovi\'c 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 ${G-v|v\in V}$ for $|V|\geq 3$? 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 ${G-e|e\in E}$? In this paper, we prove that if $|V|\neq |E|$, then the characteristic polynomial of $G$ can be reconstructed from the characteristic polynomials of all subgraphs in ${G-uv, G-u-v|uv\in E}$, 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 ${G-e|e\in E}$ (resp. if $|V|\neq |E|$).