How to count trees?

Details How to count trees? How to count trees?

2005-01-25

arxiv.org/abs/cond-mat/0501594v3

Int. J. Mod. Phys. C16 (2005) 1527

Physics

Condensed Matter

Statistical Mechanics

13 pages, 5 figures in 67 eps file, elsart

Scientific paper

10.1142/S0129183105008114

We propose a new topological invariant of unlabeled trees of N nodes. The invariant is a set of Nx2 matrices of integers, with sum_j k^{d_{i,j}} and v_i as the matrix elements, where d_{i,j} are the elements of the distance matrix and v_i denotes i-th node's degree and k in N. To compare the invariant calculated for possibly different graphs, the matrix rows are ordered with respect to first column, and -- if necessary -- with respect to the second one. We use the new invariant to evaluate from below the number of topologically different unlabeled trees up to N=17. The results slightly exceed the asymptotic evaluation of Otter.

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