On the spectral moments of trees with a given bipartition

For two given positive integers $p$ and $q$ with $p⩽q$, we denote \mathscr{T}_n^{p, q}={T: T is a tree of order $n$ with a $(p, q)-bipartition}. For a graphG$with$n$vertices,let$A(G)$ be its adjacency matrix with eigenvalues $\lambda_1(G), \lambda_2(G), …, \lambda_n(G)$innon-increasingorder\mathrm{.}Thenumber$S_k(G):=\sum_{i=1}^{n}\lambda_i^k(G),(k=0, 1, …, n-1)$iscalledthe$k$th spectral moment of $G$. Let $S(G)=(S_0(G),S_1(G),\mathrm{.}\mathrm{.}\mathrm{.},S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, one has $G_1{\prec }_s{G}_2$ if for some $k\in 1,2,\mathrm{.}\mathrm{.}\mathrm{.},n-1$, $S_i(G_1)=S_i(G_2) (i=0,1,…,k-1)$and$S_k(G_1)<S_k(G_2)$ holds. In this paper, the last four trees, in the $S$-order, among ${\mathcal{T}}_n^{p,q}(4⩽p⩽q)$ are characterized.