Spanning trees and a conjecture of Kontsevich

Kontsevich conjectured that the number of zeros over the fieldF_q of a certain polynomialQ_G associated with the spanning trees of a graphG is a polynomial function ofq. 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.Partially supported by NSF grant #DMS-9743966.