Hook immanantal equalities for linear combination matrices of (di)graphs and their applications
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916914974949376 |
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| author | Dong, Xiangshuai Wu, Tingzeng Lai, HongJian |
| author_facet | Dong, Xiangshuai Wu, Tingzeng Lai, HongJian |
| contents | Let $χ_λ$ 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 $χ_λ$ by \begin{eqnarray*} d_λ(M) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n m_{iσ(i)}. \end{eqnarray*} For $λ= (k, 1^{n-k})$, 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 $βD(G)+γA(G)$ and $βD(\overrightarrow{G})+γA(\overrightarrow{G})$, where $β$ and $γ$ are real numbers. As applications, we derive recursive formulas for the hook immanantal polynomials and hook immanants of graph matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hook immanantal equalities for linear combination matrices of (di)graphs and their applications Dong, Xiangshuai Wu, Tingzeng Lai, HongJian Combinatorics Let $χ_λ$ 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 $χ_λ$ by \begin{eqnarray*} d_λ(M) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n m_{iσ(i)}. \end{eqnarray*} For $λ= (k, 1^{n-k})$, 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 $βD(G)+γA(G)$ and $βD(\overrightarrow{G})+γA(\overrightarrow{G})$, where $β$ and $γ$ are real numbers. As applications, we derive recursive formulas for the hook immanantal polynomials and hook immanants of graph matrices. |
| title | Hook immanantal equalities for linear combination matrices of (di)graphs and their applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.17743 |