On the skew spectral moments of trees with a given bipartition

Let G be a simple graph, and G be an oriented graph of G with an orientation and skew-adjacency matrix S(G). Let λ1(G), λ2(G), . . . , λn(G) be the eigenvalues of S(G). The number n i=1 λk(G) (k = 0, 1, . . . , n − 1), denoted by Tk(G), is called the k-th skew spectral moment of G, and T (G) = (T0(G), T1(G), . . . , Tn−1(G)) is the sequence of skew spectral moments of G. Suppose G1 and G2 are two digraphs. We shall write G1 ≺T G2 (G1 comes before G2 in a T -order) if for some k (1 ≤ k ≤ n−1), Ti(G1) = Ti(G2) (i = 0, 1, . . . , k−1) and Tk(G1) Tk(G2) hold. For two given positive integers p and q with p ≤ q, we denote T 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 T p,q. Furthermore, the last three trees, in the T -order, underlying graphs among T p,q (4 ≤ p ≤ q) are characterized.