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Tree Representations of Non-symmetric Group-Valued Proximities

✍ Scribed by Charles Semple; Mike Steel

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 158 KB
Volume: 23
Category: Article
ISSN: 0196-8858
DOI: 10.1006/aama.1999.0662

No coin nor oath required. For personal study only.

✦ Synopsis

Let X be a finite set and let d be a function from X = X into an arbitrary group G G. An example of such a function arises by taking a tree T whose vertices Ž include X, assigning two elements of G G to each edge of T one for each . Ž . orientation of the edge , and setting d i, j equal to the product of the elements along the directed path from i to j. We characterize conditions when an arbitrary function d can be represented in this way, and show how such a representation may be explicitly constructed. We also describe the extent to which the underlying tree and the edge weightings are unique in such a representation. These results generalize a recent theorem involving undirected edge assignments by an Abelian group. The non-Abelian bi-directed case is of particular relevance to phylogeny reconstruction in molecular biology.

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