**Question** If graph is tree what can be said about its [adjacency matrix](http://en.wikipedia.org/wiki/Adjacency_matrix) ? And vice versa ?


Especially I am interested in case when graph is [bipartite graph](http://en.wikipedia.org/wiki/Bipartite_graph).


Such graphs are related to error-correction codes (see e.g. [Adjacency matrices of graphs as parity check matrices of error correcting codes](https://mathoverflow.net/questions/89658/adjacency-matrices-of-graphs-as-parity-check-matrices-of-error-correcting-codes) ). 
If they are trees [belief propagation](http://en.wikipedia.org/wiki/Belief_propagation) is known to produce exact results.



If graph is tree what can be said about its adjacency matrix ? 
A graph is bipartite iff the odd powers of the adjacency matrix have all 0's on the diagonal. So this implies that the sum of the $i$-th powers of the eigenvalues is 0 for each odd $i$. Since the adjacency matrix is symmetric, it has real eigenvalues. Thus, the eigenvalues are real numbers $\lambda\_1,\dots ,\lambda\_n $ with $\sum\_{j=1}^n \lambda\_j^i = 0$ for each odd $i$. 


I am guessing this probably should mean that the nonzero eigenvalues come in pairs of equal magnitude and opposite sign. I wonder if there is a good trick for efficiently proving this sort of thing -- about collections of real numbers satisfying such an infinite family of relations? (If so, I don't know this trick.)


${\bf Edit:}$ Douglas Zare proved my above conjecture as a comment, so it is true for bipartite graphs that the nonzero eigenvalues of the adjacency matrix come in pairs of equal magnitude and opposite sign. 


