Monday, March 23, 2009

Adjacency matrix of a tree..

Is there a purely algebraic characterization of the adjacency matrix of a tree ? In other words, given an n X n boolean matrix, can I determine whether it's the adjacency matrix of a tree with purely algebraic conditions (rather than writing down the induced graph and checking).

The reason I'd like this is because I want to talk about the space of such matrices, and I'd like to speak algebraically, rather than indirectly via a conversion to a graph. This seems like something that should be fairly well known: I've found some descriptions that involve all determinants of minors, and was hoping there was something more "compact".

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