# Are trees spectrally determined?

Are trees (connected acyclic graphs) determinable up to isomorphism by their spectra or characteristic polynomials? If not, what other pieces of information may help determine the tree?

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Wikipedia asserts that almost all trees are isospectral (en.wikipedia.org/wiki/Spectral_graph_theory#Isospectral_graphs). –  Qiaochu Yuan Aug 23 at 5:47
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## 2 Answers

Any tree is uniquely defined by the distances (the length of the shortest chain) between its leaves (vertices of degree 1); see:

Smolenskii Ye. A. A method for the linear recording of graphs Zh. Vychisl. Mat. Mat. Fiz., 2:2 (1962), pp.371–372.

http://www.mathnet.ru/php/person.phtml?&personid=26482&option_lang=eng

http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:03207296&type=pdf&format=complete

K. A. Zaretskii, Constructing a tree on the basis of a set of distances between the hanging vertices”. Uspekhi Mat. Nauk, 20:6(126) (1965), pp. 90–92 (Russian).

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Brendan McKay showed that there are many pairs of non-isomorphic trees with identical characteristic polynomials, as well as several other algebraic invariants. See Thm 4.2 in the linked paper.

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Schenk showed in 1973 that there were infinitely many pairs of non-isomorphic trees (reference in McKay's paper). He showed that the probability that a tree on $n$ vertices is determined by its spectrum goes to zero as $n$ goes to infinity. –  Chris Godsil Aug 23 at 11:50
Err, Schwenk showed... –  Chris Godsil Aug 23 at 14:51
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