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Spectral radius of r-uniform supertrees with perfect matchings

Frontiers of Mathematics in China 2018年第6期13卷 1489-1499页

作者： Lei ZHANG An CHANG Center for Discrete Mathematics and Theoretical Computer'Science Fuzhou University Fuzhou 350116China

supertree is a connected and acyclic hypergraph. The set of r-uniform supertrees with n vertices and the set of r-uniform supertrees with perfect matchings on rk vertices are denoted by Tn and Tr,k, respectively. H. Li, J. Shao, and L. Qi [J. Comb. Optim., 2016, 32(3): 741–764] proved that the hyperstar Sn,r attains uniquely the maximum spectral radius in Tn. Focusing on the spectral radius in Tr,k, this paper will give the maximum value in Tr,k and their corresponding supertree.

关键词： supertrees spectral radius perfect matching 15A42 05C50 05C65

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Fifty years of character compatibility concepts at work

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植物分类学报 2008年第2期46卷 109-129页

作者： George F.ESTABROOK Department of Ecology and Evolutionary Biology The University of Michigan Ann Arbor MI 4809-1048 USA

In the mid 19th century, systematic biologists realized that observable similarities and differences among a group of related species could be the basis for hypotheses about the evolutionary relationships among the species and their ancestors. Such hypotheses can be expressed as characters. A character is comprised of two or more character states of species considered to be similar with respect to a basis for comparison. The states of a character may also be arranged into a character state tree to hypothesize speciation events associated with changes from one character state to another. In the mid 20th century, some systematists realized that sometimes pairs of characters （or character state trees） could be incompatible as hypotheses, i.e., they could not both be true. Through the 1950s, ＇60s and ＇70s, tests for, and ways to resolve, incompatibilities were used to estimate an ancestor rela-tion based on mutually compatible characters. An estimate was often shown as a diagram connecting ancestors to their immediate descendants （not quite correctly） called a phylogenetic tree. More recently, other applications of compatibility concepts have been developed, including： identify characters that appear to be random in the context of their data set; combine estimates of ancestor relations for subsets of taxa in a larger collection into a single estimate （a so-called supertree） for the whole collection; and interpret geographic patterns in an evolutionary context.

关键词： biogeography character compatibility character evaluation convex groups phylogenetic trees,reciprocal illumination speciation supertrees

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