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.
Categories: math.CO