On the edge reconstruction of the second immanantal polynomials of undirected graph and digraph

TL;DR

This paper demonstrates that the second immanantal polynomial of a graph's adjacency or Laplacian matrix can be reconstructed from the second immanantal polynomials of all subgraphs, advancing graph reconstruction theory.

Contribution

It introduces methods to reconstruct second immanantal polynomials of graphs and digraphs from subgraph polynomials, extending graph reconstruction results to these polynomials.

Findings

Reconstruction of $d_{2}(xI-A(G))$ from subgraph polynomials.

Reconstruction of $d_{2}(xI-A( ightarrow G))$ from subgraph polynomials.

Reconstruction of $d_{2}(xI-D( ightarrow G))$ and related polynomials from subgraph Laplacians.

Abstract

Let $M=(m_{ij})$ be an $n\times n$ matrix. The second immanant of matrix $M$ is defined by \begin{eqnarray*} d_{2}(M)=\sum_{\sigma\in S_{n}}\chi_{2}(\sigma)\prod_{s=1}^{n}m_{s\sigma(s)}, \end{eqnarray*} where $\chi_{2}$ is the irreducible character of $S_{n}$ corresponding to the partition $(2^{1},1^{n-2})$. The polynomial $d_{2}(xI-M)$ is called the second immanantal polynomial of matrix $M$. Denote by $D(G)$ (resp. $D(\overrightarrow{G})$) and $A(G)$ (resp. $A(\overrightarrow{G})$) the diagonal matrix of vertex degrees and the adjacency matrix of undirected graph $G$ (resp. digraph $\overrightarrow{G}$), respectively. In this article, we prove that $d_{2}(xI-A(G))$ (resp. $d_{2}(xI-A(\overrightarrow{G}))$) can be reconstructed from the second immanantal polynomials of the adjacency matrix of all subgraphs in $\{G-uv,G-u-v|uv\in E(G)\}$ (resp. $\{\overrightarrow{G}-e|e\in E(\overrightarrow{G})\}$). Furthermore, the polynomial $d_{2}(xI-D(\overrightarrow{G})\pm A(\overrightarrow{G}))$ can also be reconstructed by the second immanantal polynomials of the (signless) Laplacian matrixs of all subgraphs in $\{\overrightarrow{G}-e|e\in E(\overrightarrow{G})\}$, respectively.