几乎永久谱同构图
Nearly permanental cospectral graphs
AI总结：
本文证明了Lv等人提出的行列式相关几乎谱同构图的模4结果对永久值相关情形也成立，并在ℱ₂[x]中对所有不可约特征标建立了几乎不变量谱同构图的类似结果。
AI中文摘要：
设G为n阶简单图，其邻接矩阵为A=(aᵢⱼ)。矩阵A的行列式（detA）和永久值（perA）分别定义为：detA=∑_{σ∈Sₙ} sgn(σ)∏_{i=1}^n a_{iσ(i)}，perA=∑_{σ∈Sₙ} ∏_{i=1}^n a_{iσ(i)}。多项式φ(G;x)=det(xI-A(G))和π(G;x)=per(xI-A(G))分别称为G的特征多项式和永久多项式。若两个图关于行列式（或永久值）的特征多项式（或永久多项式）之差为常数，则称它们关于行列式（或永久值）几乎谱同构。Lv等人提出了关于行列式的几乎谱同构图问题，并给出了模4情形下的部分结果。本文主要证明，对应的结果对于关于永久值的几乎谱同构图问题同样成立。行列式和永久值分别是对称群Sₙ的不可约特征标(1ⁿ)和(n)对应的不变量（immanant）。此处，矩阵A的不变量d_λ(A)定义为：d_λ(A)=∑_{σ∈Sₙ} χ_λ(σ)∏_{i=1}^n a_{iσ(i)}，其中χ_λ是由分划λ索引的Sₙ的不可约特征标。与χ_λ关联的G的不变量多项式为φ_λ(G;x)=d_λ(xI-A)。本文还对所有不可约特征标χ_λ，在ℱ₂[x]中建立了关于几乎不变量谱同构图的类似结果。
英文摘要：
Let $G$ be a simple graph of order $n$ with adjacency matrix $A= (a_{ij})$. The \emph{determinant} and the \emph{permanen}t of the matrix $A$ are defined as \[\mathrm{det}A= \sum_{σ\in S_n}\mathrm{sgn}(σ) \prod_{i=1}^n a_{iσ(i)}\quad\text{and}\quad\mathrm{per}A= \sum_{σ\in S_n} \prod_{i=1}^n a_{iσ(i)},\]respectively. The polynomials $ϕ(G;x) =\mathrm{det}(xI-A(G))$ and $π(G;x) =\mathrm{per}(xI-A(G))$ are called the \emph{characteristic polynomial} and the \emph{permanental polynomial} of $G$, respectively. Two graphs are said to be \emph{nearly cospectral} with respect to the determinant (resp. permanent) if the difference of their characteristic (resp. permanental) polynomials is a constant. Lv et al. introduced the nearly cospectral graphs problem with respect to the determinant, and provided partial results in the case modulo 4. In this paper, we mainly prove that the corresponding results also hold for the nearly cospectral graphs problem with respect to the permanent. The determinant and permanent are the immanants corresponding to the irreducible characters $(1^n)$ and $(n)$ of the symmetric group $ S_n $, respectively. Here, the \emph{immanant} $d_λ(A)$ of $A$ is defined as \[d_λ(A) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n a_{iσ(i)},\] where $χ_λ$ is the irreducible character of $ S_n $ indexed by the partition $ λ$. The immanantal polynomial of $G$ associated with $ χ_λ$ is given by $ ϕ_λ(G;x)=d_λ(xI-A) $. In this paper, we also establish a similar result for nearly immanantal cospectral graphs in $\mathbb{F}_2[x]$ for all irreducible characters $χ_λ$.
