arXiv subjects

Xiangshuai Dong

Publications and source records attributed to Xiangshuai Dong.

5 recordsLinked to original sources

arXiv2026

On the nonnegativity of monomial immanants for hook partitions

Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $λ$ be a partition of $n$. Let $ϕ^λ$ be the class function dual to the Young permutation character, and let $$ ϕ^λ[A] = \sum_{σ\in\mathfrak{S}_n}ϕ^λ(σ) \prod_{i=1}^{n}a_{iσ(i)} $$ be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of $A$ of order at most $r$ are nonnegative, and the partition $λ$ has length at most $r$, is it true that $ϕ^λ[A]\ge 0$? This paper solves the problem for all hook partitions.

math.CO↗

arXiv2026

Solution to an open problem on the computational complexity of immanant

Immanants are a class of generalized matrix functions associated with the irreducible characters of the symmetric group. Bürgisser [SIAM J. Comput., 30 (2000), pp. 1023--1040] proved that the computation of hook immanants and immanants corresponding to rectangular Young diagrams of polynomially growing width is VNP-complete under $p$-projections. And he posed an open problem: whether the family of immanants corresponding to rectangular Young diagrams of width $2$ is VNP-complete under $p$-projections. This paper gives a solution to this problem. We prove that, over any field of characteristic zero, the immanant families associated with rectangular Young diagrams of width $2$ and of length $2$ are both VNP-complete under $p$-projections.

math.CO↗

arXiv2026

Immanantal polynomials of the linear combination matrices of graphs

In this paper, we focus on the study of immanantal polynomials for linear combination matrices composed of the degree matrix and adjacency matrix of a graph. First, applying the concept of vertex orientation for general graphs, we provide a combinatorial interpretation of the coefficients of the immanantal polynomials for the linear combination matrices of graphs, and we also characterize the bounds of these coefficients. These bounds implicitly encompass the existing results of Chan and Lam on trees and bipartite graphs. Furthermore, we give a solution to the open problem posed by Merris. Second, we characterize the first six coefficients of the hook immanantal polynomial. And the necessary and sufficient condition under which the linear combination matrices of two regular graphs have the same hook immanantal polynomial is proved. Third, we generalize the Frobenius--König theorem and the Laplace expansion theorem to immanants. Using these two theorems, we show that the star degree of a graph is always a lower bound for the multiplicity of a certain root of the immanantal polynomial of its linear combination matrix. Finally, we derive formulas for the first six coefficients of the hook immanantal polynomial for several important graph matrices.

math.CO↗

arXiv2025

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

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.

math.CO↗

arXiv2025

Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices

Let $\mathscr{U}(n,τ)$ be the set of all {\rm(0,1)}-matrices of order $n$ with exactly $τ$ 0's. Brualdi et al. investigated the maximum permanents of all matrices in $\mathscr{U}(n,τ)$(R.A. Brualdi, J.L. Goldwasser, T.S. Michael, Maximum permanents of matrices of zeros and ones, J. Combin. Theory Ser. A 47 (1988) 207--245.). And they put forward an open problem to characterize the maximum permanents among all matrices in $\mathscr{U}(n,τ)$. In this paper, we focus on the problem. And we characterize the maximum permanents of all matrices in $\mathscr{U}(n,τ)$ when $n^{2}-3n\leqτ\leq n^{2}-2n-1$. Furthermore, we also prove the maximum permanents of all matrices in $\mathscr{U}(n,τ)$ when $σ-kn\equiv0 (mod~k+1)$ and $(k+1)n-σ\equiv0(mod~k)$, where $σ=n^{2}-τ$, $kn\leqσ\leq (k+1)n$ and $k$ is integer.

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