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

Let $M=(m_{ij})$ be an $n\times n$ matrix. The second immanant of matrix $M$ is defined by $d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{s=1}^{n}m_{s\sigma(s)}$, 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 the matrix $M$. Denote by $D(G)$ (resp. $D(\overrightarrow{G})$) and $A(G)$ (resp. $A(\overrightarrow{G})$) the diagonal matrix of vertex degrees (resp. vertex in-degrees) and the adjacency matrix of 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 matrices of all subgraphs in $\{G-uv,G-u-v\mid uv\in E(G)\}$ (resp. $\{\overrightarrow{G}-e\mid e\in E(\overrightarrow{G})\}$). Furthermore, the polynomial $d_2(xI-D(\overrightarrow{G})\pm A(\overrightarrow{G}))$ can also be reconstructed by the second immanantal polynomials of the Laplacian (signless Laplacian) matrices of all subgraphs in $\{\overrightarrow{G}-e\mid e\in E(\overrightarrow{G})\}$.