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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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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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    $\begingroup$ 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. $\endgroup$ – Chris Godsil Aug 23 '13 at 11:50
  • $\begingroup$ Err, Schwenk showed... $\endgroup$ – Chris Godsil Aug 23 '13 at 14:51
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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).

http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=6134&option_lang=eng http://www.zentralblatt-math.org/zmath/search/?an=Zbl%200151.33302

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