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On the ordering of trees by the Laplacian coefficients | | Aleksandar Ili' c
; | | Date: |
21 Apr 2011 | | Abstract: | We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen,
extit{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. | | Source: | arXiv, 1104.4280 | | Services: | Forum | Review | PDF | Favorites | |
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