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

TL;DR

This paper investigates the edge and vertex reconstruction of characteristic and permanental polynomials of simple graphs, providing new conditions under which these polynomials can be uniquely reconstructed from subgraph polynomials.

Contribution

The paper proves that the characteristic and permanental polynomials of a graph can be reconstructed from subgraphs' polynomials when the number of vertices differs from the number of edges, extending reconstruction results.

Findings

Reconstruction of characteristic polynomial from subgraphs when |V| ≠ |E|.

Reconstruction of permanental polynomial under similar conditions.

Reconstruction of Laplacian and signless Laplacian polynomials from edge-deleted subgraphs.

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 $|V|\neq |E|$).