Hook immanantal equalities for linear combination matrices of (di)graphs and their applications

Let $\chi_\lambda$ be an irreducible character of the symmetric group $S_n$. For an $n \times n$ matrix $M = (m_{ij})$, define the immanant of $M$ corresponding to $\chi_\lambda$ by \begin{eqnarray*} d_\lambda(M) = \sum_{\sigma \in S_n} \chi_\lambda(\sigma) \prod_{i=1}^n m_{i\sigma(i)}. \end{eqnarray*} For $\lambda = (k, 1^{n-k})$, the immanant $d_{(k, 1^{n-k})}(M)$ is called the hook immanant and denoted by $d_k(M)$. The hook immanant polynomial of matrix $M$ is defined as $d_{k}(xI_n - M)$, where $I_n$ is the $n \times n$ identity matrix. Let $G$ and $\overrightarrow{G}$ be a graph and a digraph, respectively. Suppose that $D(G)$ and $A(G)$ (resp. $D(\overrightarrow{G})$ and $A(\overrightarrow{G})$) are the degree matrix and adjacency matrix of $G$ (resp. $\overrightarrow{G}$), respectively. In this paper, we characterize two hook immanantal equalities for the linear combination of matrices $\beta D(G)+\gamma A(G)$ and $\beta D(\overrightarrow{G})+\gamma A(\overrightarrow{G})$, where $\beta$ and $\gamma$ are real numbers. As applications, we derive recursive formulas for the hook immanantal polynomials and hook immanants of graph matrices.