Tree representations of non-symmetric group-valued proximities

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Discussion / Working Papers
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University of Canterbury
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Date
1999
Authors
Semple, Charles
Steel, M.
Abstract

Let X be a finite set and let d be a function from X x X into an arbitrary group Q. An example of such a function arises by taking a tree T whose vertices include X, assigning two elements of Q to each edge of T ( one for each 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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ANZSRC fields of research
Fields of Research::49 - Mathematical sciences::4901 - Applied mathematics::490102 - Biological mathematics
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