Nearly permanental cospectral graphs

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 $χ_λ$.