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On the ordering of trees by the Laplacian coefficients

We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, \textit{Ordering trees by the Laplacian coefficients}, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018] on the partial ordering of trees with given diameter. For two $n$-vertex trees $T_1$ and $T_2$, if $c_k (T_1) \leqslant c_k (T_2)$ holds for all Laplacian coefficients $c_k$, $k = 0, 1, ..., n$, we say that $T_1$ is dominated by $T_2$ and write $T_1 \preceq_c T_2$. We proved that among $n$ vertex trees with fixed diameter $d$, the caterpillar $C_{n, d}$ has minimal Laplacian coefficients $c_k$, $k = 0, 1,..., n$. The number of incomparable pairs of trees on $\leqslant 18$ vertices is presented, as well as infinite families of examples for two other partial orderings of trees, recently proposed by Mohar. For every integer $n$, we construct a chain $\{T_i\}_{i = 0}^m$ of $n$-vertex trees of length $\frac{n^2}{4}$, such that $T_0 \cong S_n$, $T_m \cong P_n$ and $T_i \preceq_c T_{i + 1}$ for all $i = 0, 1,..., m - 1$. In addition, the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths. We determine the relations among the extremal trees with fixed maximum degree, and with perfect matching and further support the Laplacian coefficients as a measure of branching.