论文标题
laplacian矩阵，用于极平衡和不平衡的系统发育树
Laplacian Matrices for Extremely Balanced and Unbalanced Phylogenetic Trees
论文作者
论文摘要
系统发育树是研究物种之间进化关系的重要工具。诸如sackin，colless和总抗烯的指数之类的措施已被广泛用于量化树木平衡，这是系统发育的一个关键特性。最近，基于与树相关的拉普拉斯矩阵的光谱，引入了一项新的建议。在这项工作中，我们分析了两个极端情况，对Laplacian矩阵计算，对应于完全平衡和完全不平衡的树木。对于最大平衡的树木，衍生了Laplacian基质的封闭形式，但我们提出了一种构建算法。我们表明，完全平衡的树木的拉普拉斯矩阵显示自相似的模式，导致高度退化的特征值。退化是该拓扑结构的主要签名，因为它在完全不平衡的树木中完全不存在。我们还为这些拓扑结构的Laplacian矩阵最大的特征值建立了一些分析和数值结果。
Phylogenetic trees are important tools in the study of evolutionary relationships between species. Measures such as the index of Sackin, Colless, and Total Cophenetic have been extensively used to quantify tree balance, one key property of phylogenies. Recently a new proposal has been introduced, based on the spectrum of the Laplacian matrix associated with the tree. In this work, we calculate the Laplacian matrix analytically for two extreme cases, corresponding to fully balanced and fully unbalanced trees. For maximally balanced trees no closed form for the Laplacian matrix was derived, but we present an algorithm to construct it. We show that Laplacian matrices of fully balanced trees display self-similar patterns that result in highly degenerated eigenvalues. Degeneracy is the main signature of this topology, since it is totally absent in fully unbalanced trees. We also establish some analytical and numerical results about the largest eigenvalue of Laplacian matrices for these topologies.
