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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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Wikipedia asserts that almost all trees are isospectral ( – Qiaochu Yuan Aug 23 '13 at 5:47

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.

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 '13 at 11:50
Err, Schwenk showed... – Chris Godsil Aug 23 '13 at 14:51

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