**Question** If graph is tree what can be said about its [adjacency matrix][1] ?  And vice versa ?

Especially I am interested in case  when graph is [bipartite graph][2].

Such graphs are related to error-correction codes (see e.g. http://mathoverflow.net/questions/89658/adjacency-matrices-of-graphs-as-parity-check-matrices-of-error-correcting-codes). 
If they are trees [belief propagation][3] is known to produce exact results.


  [1]: http://en.wikipedia.org/wiki/Adjacency_matrix
  [2]: http://en.wikipedia.org/wiki/Bipartite_graph
  [3]: http://en.wikipedia.org/wiki/Belief_propagation