dorsal/arxiv
View SchemaUbiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials
| Authors | Frederick A. Matsen, Steven N. Evans |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0512010 |
| URL | https://arxiv.org/abs/q-bio/0512010 |
Abstract
There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such representations involve a specific labeling of the vertices or at least the leaves, and so it is natural to attempt to identify trees by some feature of the associated matrices that is invariant under relabeling. An obvious candidate is the spectrum of eigenvalues (or, equivalently, the characteristic polynomial). We show for any of these choices of matrix that the fraction of binary trees with a unique spectrum goes to zero as the number of leaves goes to infinity. We investigate the rate of convergence of the above fraction to zero using numerical methods. For the adjacency and Laplacian matrices, we show that that the {\em a priori} more informative immanantal polynomials have no greater power to distinguish between trees.
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"abstract": "There are several common ways to encode a tree as a matrix, such as the\nadjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of\nthe natural random walk), and the matrix of pairwise distances between leaves.\nSuch representations involve a specific labeling of the vertices or at least\nthe leaves, and so it is natural to attempt to identify trees by some feature\nof the associated matrices that is invariant under relabeling. An obvious\ncandidate is the spectrum of eigenvalues (or, equivalently, the characteristic\npolynomial). We show for any of these choices of matrix that the fraction of\nbinary trees with a unique spectrum goes to zero as the number of leaves goes\nto infinity. We investigate the rate of convergence of the above fraction to\nzero using numerical methods. For the adjacency and Laplacian matrices, we show\nthat that the {\\em a priori} more informative immanantal polynomials have no\ngreater power to distinguish between trees.",
"arxiv_id": "q-bio/0512010",
"authors": [
"Frederick A. Matsen",
"Steven N. Evans"
],
"categories": [
"q-bio.PE"
],
"title": "Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials",
"url": "https://arxiv.org/abs/q-bio/0512010"
},
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"execution_id": "e71198a9-65ba-4606-8376-9035ccbaf812",
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