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Usually one constructs a graph and then asks questions about the adjacency matrix's (or some close relative like the Laplacian) eigenvalue decomposition (also called the spectra of a graph).

But what about the reverse problem? Given $n$ eigenvalues, can one (efficiently) find a graph that has this spectra?

I suspect that in general this is hard to do (and might be equivalent to GI) but what if you relax some of the conditions a bit? What if you make conditions that there is no multiplicity of eigenvalues? What about allowing graphs that have "close" spectra by some distance metric?

Any references or ideas would be welcome.

EDIT:

As Suresh points out, if you allow undirected weighted graphs with self loops, this problem becomes pretty trivial. I was hoping to get answers on the set of undirected, unweighted simple graphs but I would be happy with simple unweighted directed graphs as well.

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I think you might need to modify the question to 'unweighted undirected graphs with no self loops' or something like that ? I can imagine constructing a diagonal matrix with the required eigenvalues and declaring it to be a disconnected graph with weighted selfloops ? –  Suresh Venkat Dec 11 '10 at 19:47
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Even simpler question (I don't know the answer) is how to construct simple connected graphs whose top few eigenvalues are given –  Yaroslav Bulatov Dec 12 '10 at 1:07
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An alternative way of stating the question (the version with simple undirected graphs) is: given n real numbers (in some format), decide whether there exists an n×n symmetric 0/1 matrix with zero diagonals such that its n eigenvalues are the given numbers. –  Tsuyoshi Ito Dec 12 '10 at 1:12
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@Yaroslav: I am not sure, but that problem sounds harder to me than the case where all n eigenvalues are given. –  Tsuyoshi Ito Dec 12 '10 at 12:57
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Tiny observation: If we have no restrictions on the eigenvalues, the problem is really hard (even not include the algorithmic part) since this will implies the (non-)existence to the 57-regular Moore graph, which the eigenvalues are all known. –  Hsien-Chih Chang 張顯之 Dec 12 '10 at 17:41

3 Answers 3

Cvetcovic et all in Section 3.3 of "Recent results in the theory of graph spectra" goes over algorithms for constructing graphs given spectrum in some special cases

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Even asking weather a graph with given spectra exist is a tough question. This is witnessed by the open problem of determining whether there exist a graph of girth 5 diameter 2 and order 3250 whose spectrum (if it exists) is known.

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One other obstacle in defining your question is that the are isospectral (same eigenvalues) but non-isomorphic graphs. So given a list of eigenvalues in such a case, which graph do you want? Maybe you just want an algorithm to return one random element of the set of such non-isomorphic graphs?

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I was thinking something along the lines of sampling from the space of graphs that are isospectral but this seems like we're quickly descending into a GI equivalent problem (thus my comment above). For simplicity, we could restrict to all distinct eigenvalues (which, if IIC ensures a unique graph) but I'm really just trying to see what's known or out there. –  user834 Dec 12 '10 at 21:17
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I don't think distinct eigenvalues ensures reconstructibility, here are some spectra of isospectral graphs over 7 nodes yaroslavvb.com/upload/save/cstheory-isospectral.png –  Yaroslav Bulatov Dec 12 '10 at 22:57
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I like the random element formulation. I would be interested to know if it's equivalent to GI. One reason I'm interested in the random element formulation, is the question, raised in my paper with Arora and Steurer on unique games, of whether graphs with certain spectrum can be expanders for small sets. Intuitively, one may hope that a random graph with this spectrum will be the best expander possible for all set sizes, and so insight on reverse spectra may be useful there. –  Boaz Barak Dec 13 '10 at 2:37
    
@Yaroslav: Thank you for that link and thanks for correcting me! –  user834 Dec 13 '10 at 20:09
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@user834: Re: your comment about a GI-equivalent problem. Note that determining isomorphism of graphs with bounded eigenvalue multiplicity (in particular, graphs with no multiple eigenvalues) can be done in polynomial time. –  Joshua Grochow Oct 30 '13 at 17:58

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