### Annals of Combinatorics 2(1998) 351-363 Spanning Trees and a Conjecture of Kontsevich
Richard P. Stanley Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
rstan@math.mit.edu Received October 30, 1998 **AMS Subject Classification**: 05E99
**Abstract. **Kontsevich conjectured that the number of zeros over the field F_{q} of a certain polynomial Q_{G} associated with the spanning trees of a graph G is a polynomial function of q. We show the connection between this conjecture, the Matrix--Tree Theorem, and orthogonal geometry. We verify the conjecture in certain cases, such as the complete graph, and discuss some modifications and extensions.
**Keywords**: spanning tree, Matrix--Tree Theorem, orthogonal geometry
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