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_۱(\vec{G}), \lambda_۲(\vec{G}),\ldots,\lambda_n(\vec{G}) be the eigenvalues of S(\vec{G}). The number \sum_{i=۱}^{n}\lambda_i^k(\vec{G}) (k=۰, ۱,\ldots,n-۱), denoted by T_k(\vec{G}), is called the k-th {\em skew spectral moment} of \vec{G}, and T(\vec{G})=(T_۰(\vec{G}),T_۱(\vec{G}),\ldots, T_{n-۱}(\vec{G})) is the sequence of skew spectral moments of \vec{G}. Suppose \vec{G}_۱ and \vec{G}_۲ are two digraphs. We shall write \vec{G}_۱\prec_T \vec{G}_۲ (\vec{G}_۱ comes before \vec{G}_۲ in a T-order) if for some k (۱ \leq k \leq n-۱), T_i(\vec{G}_۱)=T_i(\vec{G}_۲) (i=۰, ۱,\ldots,k-۱) and T_k(\vec{G_۱})< T_k(\vec{G}_۲) 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}~(۴\leq p\leq q) are characterized.