Edge reconstruction for second immanantal Laplacian polynomials of undirected graphs

Determine whether the second immanantal polynomials of the Laplacian and signless Laplacian matrices of an undirected graph satisfy the edge reconstruction conjecture.

Background

The paper establishes edge-reconstruction formulas for the second immanantal polynomial of the adjacency matrix of an undirected graph when the numbers of vertices and edges are unequal. It also proves analogous reconstruction results for the adjacency, Laplacian, and signless Laplacian second immanantal polynomials of digraphs. The authors leave unresolved whether the corresponding second immanantal polynomials of the Laplacian and signless Laplacian matrices of undirected graphs can be reconstructed from the relevant edge-deck data, as required by the edge reconstruction conjecture.