On the skew spectral moments of trees with a given bipartition

Let $G$ be a simple graph, and $\vec{G}$ be an oriented graph of $G$ with an orientation and skew-adjacency matrix $S(\vec{G})$. Let $\lambda_1(\vec{G}), \lambda_2(\vec{G}),\ldots,\lambda_n(\vec{G})$ be the eigenvalues of $S(\vec{G})$. The number $\sum_{i=1}^{n}\lambda_i^k(\vec{G})$ $(k=0, 1,\ldots,n-1)$, denoted by $T_k(\vec{G})$, is called the $k$-th {\em skew spectral moment} of $\vec{G}$, and $T(\vec{G})=(T_0(\vec{G}),T_1(\vec{G}),\ldots,$ $T_{n-1}(\vec{G}))$ is the sequence of skew spectral moments of $\vec{G}$. Suppose $\vec{G}_1$ and $\vec{G}_2$ are two digraphs. We shall write $\vec{G}_1\prec_T \vec{G}_2$ ($\vec{G}_1$ comes before $\vec{G}_2$ in a $T$-order) if for some $k$ $(1 \leq k \leq n-1)$, $T_i(\vec{G}_1)=T_i(\vec{G}_2)$ ($i=0, 1,\ldots,k-1$) and $T_k(\vec{G_1})< T_k(\vec{G}_2)$ hold. For two given positive integers $p$ and $q$ with $p \leq q$, we denote $\mathscr T_{n}^{p,q}=\{T: T$ is a tree of order $n$ with a $(p,q)$-bipartition $\}$. In this paper, we discuss $T$-order among all trees in $\mathscr T_{n}^{p,q}$. Furthermore, the last three trees, in the $T$-order, underlying graphs among $\mathscr T_{n}^{p,q}~(4\leq p\leq q)$ are characterized.