On the Edge Reconstruction of the Second Immanantal Polynomials of Graphs and Digraphs

Let M=(mᵢⱼ) be an n× n matrix. The second immanant of matrix M is defined by d₂(M)=∑σ∈ Sₙχ₂(σ)∏ₛ₌₁ⁿmsσ(s), where χ₂ is the irreducible character of Sₙ corresponding to the partition (2¹,1ⁿ⁻²). The polynomial d₂(xI-M) is called the second immanantal polynomial of the matrix M. Denote by $D(G)$ (resp. D(G)) and $A(G)$ (resp. A(G)) the diagonal matrix of vertex degrees (resp. vertex in-degrees) and the adjacency matrix of graph G (resp. digraph G), respectively. In this article, we prove that d₂(xI-A(G)) (resp. d₂(xI-A(G))) can be reconstructed from the second immanantal polynomials of the adjacency matrices of all subgraphs in -uv,G-u-v uv∈ E(G)\ (resp. -e e∈ E(G)\). Furthermore, the polynomial d₂(xI-D(G)± A(G)) can also be reconstructed by the second immanantal polynomials of the Laplacian (signless Laplacian) matrices of all subgraphs in -e e∈ E(G)\.